Knot homology groups from instantons

P. B. Kronheimer and T. S. Mrowka

Harvard University, Cambridge MA 02138Massachusetts Institute of Technology, Cambridge MA 02139

1 Introduction

1.1 An observation of some coincidences

For a knot or link K in S3, the Khovanov homology Kh.K/ is a bigradedabelian group whose construction can be described in entirely combinatorialterms [15]. If we forget the bigrading, then as abelian groups we have, forexample,

Kh.unknot/ D Z2

andKh.trefoil/ D Z4 ˚ Z=2:

The second equality holds for both the right- and left-handed trefoils, thoughthe bigrading would distinguish these two cases.

The present paper was motivated in large part by the observation that thegroup Z4 ˚ Z=2 arises in a different context. Pick a basepoint y0 in the com-plement of the knot or link, and consider the space of all homomorphisms

� W �1�S3nK; y0

�! SU.2/

satisfying the additional constraint that

�.m/ is conjugate to��i 0

0 i

�(1)

for every m in the conjugacy class of a meridian of the link. (There is one suchconjugacy class for each component of K, once the components are oriented.The orientation does not matter here, because the above element of SU.2/ isconjugate to its inverse.) Let us write

R.K/ � Hom��1�S3nK; y0

�;SU.2/

�

2

for the set of these homomorphisms. Note that we are not defining R.K/ as aset of equivalence classes of such homomorphisms under the action of conju-gation by SU.2/. The observation, then, is the following:

Observation 1.1. In the case that K is either the unknot or the trefoil, the Kho-vanov homology of K is isomorphic to the ordinary homology of R.K/, as anabelian group. That is,

Kh.K/ Š H�.R.K//:

This observation extends to all the torus knots of type .2; p/.

To understand this observation, we can begin with the case of the unknot,where the fundamental group of the complement is Z. After choosing a gen-erator, we have a correspondence between R.unknot/ and the conjugacy classof the distinguished element of SU.2/ in (1) above. This conjugacy class is a2-sphere in SU.2/, so we can write

R.unknot/ D S2:

For a non-trivial knot K, we always have homomorphisms � which factorthrough the abelianization H1.S3nK/ D Z, and these are again parametrizedby S2. Every other homomorphism has stabilizer f˙1g � SU.2/ underthe action by conjugation, so its equivalence class contributes a copy ofSU.2/=f˙1g D RP3 to R.K/. In the case of the trefoil, for example, thereis exactly one such conjugacy class, and so

R.trefoil/ D S2 q RP3I

and the homology of this space is indeed Z4 ˚ Z=2, just like the Khovanovhomology. This explains why the observation holds for the trefoil, and the caseof the .2; p/ torus knots is much the same: for larger odd p, there are .p�1/=2copies of RP3 in the R.K/. In unpublished work, the above observation hasbeen shown to extend to all 2-bridge knots by Sam Lewallen [25].

The homology of the space R.K/, while it is certainly an invariant of theknot or link, should not be expected to behave well or share any of the more in-teresting properties of Khovanov homology; no should the coincidence notedabove be expected to hold. A better way to proceed is instead to imitate theconstruction of Floer’s instanton homology for 3-manifolds, by constructinga framework in which R.K/ appears as the set of critical points of a Chern-Simons functional on a space of SU.2/ connections on the complement of thelink. One should then construct the Morse homology of this Chern-Simons

3

invariant. In this way, one should associate a finitely-generated abelian grouptoK that would coincide with the ordinary homology ofR.K/ in the very sim-plest cases. The main purpose of the present paper is to carry through thisconstruction. The invariant that comes out of this construction is certainly notisomorphic to Khovanov homology for all knots; but it does share some of itsformal properties. The definition that we propose is a variant of the orbifoldFloer homology considered by Collin and Steer in [5].

In some generality, given a knot or link K in a 3-manifold Y , we willproduce an “instanton Floer homology group” that is an invariant of .Y;K/.These groups will be functorial for oriented cobordisms of pairs. Rather thanwork only with SU.2/, we will work of much of this paper with a more generalcompact Lie group G, though in the end it is only for the case of SU.N / thatwe are able to construct these invariants.

1.2 Summary of results

The basic construction. Let Y be a closed oriented 3-manifold, let K � Y bean oriented link, and let P ! Y be a principal U.2/-bundle. LetK1; : : : ; Kr bethe components ofK. We will say that .Y;K/ andP satisfies the non-integralitycondition if none of the 2r rational cohomology classes

12c1.P /˙

14

P:D:ŒK1�˙ � � � ˙ 14

P:D:ŒKr � (2)

is an integer class. When the non-integrality condition holds, we will define afinitely-generated abelian group I�.Y;K;P /. This group has a canonical Z=2grading, and a relative grading by Z=4.

In the case that K is empty, the group I�.Y; P / coincides with the familiarvariant of Floer’s instanton homology arising from a U.2/ bundle P ! Y withodd first Chern class [8]. We recall, in outline, how this group is constructed.One considers the space C.Y; P / of all connections in the SO.3/ bundle ad.P /.This affine space is acted on by the “determinant-1 gauge group”: the groupG .Y; P / of automorphisms of P that have determinant 1 everywhere. InsideC.Y; P / one has the flat connections: these can be characterized as criticalpoints of the Chern-Simons functional,

CS W C.Y; P /! R:

The Chern-Simons functional descends to a circle-valued function on the quo-tient space

B.Y; P / D C.Y; P /=G .Y; P /:

4

The image of the set of critical points in B.Y; P / is compact, and after per-turbing CS carefully by a term that is invariant under G .Y; P /, one obtainsa function whose set of critical points has finite image in this quotient. If.1=2/c1.P / is not an integral class and the perturbation is small, then the crit-ical points in C.Y; P / are all irreducible connections. One can arrange alsoa Morse-type non-degeneracy condition: the Hessian if CS can be assumedto be non-degenerate in the directions normal to the gauge orbits. The groupI�.Y; P / is then constructed as the Morse homology of the circle-valued Morsefunction on B.Y; P /.

In the case thatK is non-empty, the construction of I�.Y;K;P /mimics thestandard construction very closely. The difference is that we start not with thespace C of all smooth connections in ad.P /, but with a space C.Y;K;P / ofconnections in the restriction of ad.P / to Y nK which have a singularity alongK. This space is acted on by a group G .Y;K;P / of determinant-one gaugetransformations, and we have a quotient space

B.Y;K;P / D C.Y;K;P /=G .Y;K;P /:

In the case that c1.P / D 0, the singularity is such that the flat connections inthe quotient space B.Y;K;P / correspond to conjugacy classes of homomor-phisms from the fundamental group of Y nK to SU.2/which have the behavior(1) for meridians of the link. Thus, if we write C.Y;K;P / � B.Y;K;P / forthis set of critical points of the Chern-Simons functional, then we have

C.Y;K;P / D R.Y;K/=SU.2/ (3)

where R.Y;K/ is the set of homomorphisms � W �1.Y nK/! SU.2/ satisfying(1) and SU.2/ is acting by conjugation. The non-integrality of the classes (2)is required in order to ensure that there will be no reducible flat connections.

Application to classical knots. Because of the non-integrality requirement, theconstruction of I� cannot be applied directly when the 3-manifold Y has firstBetti number zero. In particular, we cannot apply this construction to “classi-cal knots” (knots in S3). However, there is a simple device we can apply. Picka point y0 in Y nK, and form the connected sum at y0 of Y and T 3, to obtaina new pair .Y#T 3; K/. Let P0 be the trivial U.2/ bundle on Y , and let Q bethe U.2/ bundle on T 3 D S1 � T 2 whose first Chern class is Poincare dual toS1 � fpointg. We can form a bundle P0#Q over Y#T 3. This bundle satisfiesthe non-integral condition, so we define

FI�.Y;K/ D I�.Y#T 3; K; P0#Q/:

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and call this the framed instanton homology of the pair .Y;K/. In the specialcase that Y D S3, we write

FI�.K/ D FI�.S3; K/:

To get a feel for FI�.Y;K/ for knots in S3, and to understand the reasonfor the word “framed” here, it is first necessary to understand that the adjointbundle ad.Q/ ! T 3 admits only irreducible flat connections, and that theseform two orbits under the determinant-one gauge group. (See section 3.1. Un-der the full gauge group of all automorphisms of ad.Q/, they form a singleorbit.) When we form a connected sum, the fundamental group becomes afree product, and we have a general relationship of the form

R.Y0#Y1/=SU.2/ D R.Y0/ �SU.2/ R.Y1/: (4)

Applying this to the connected sum .Y#T 3; K/ and recalling (3), we find thatthe flat connections in the quotient space B.Y#T 3; K; P0#Q/ form two dis-joint copies of the space we called R.Y;K/ above: that is,

C.Y#T 3; K; P0#Q/ D R.Y;K/qR.Y;K/:

Note that, on the right-hand side, we no longer have the quotient of R.Y;K/by SU.2/ as we did before at (3). The space R.Y;K/ can be thought of asparametrizing isomorphism classes of flat connections on Y nK with a framingat the basepoint y0: that is, an isomorphism of the fiber at y0 with U.2/.

For a general knot, as long as the set of critical points is non-degeneratein the Morse-Bott sense, there will be a spectral sequence starting at the ho-mology of the critical set, C, and abutting to the framed instanton homology.In the case of the unknot in S3, the spectral sequence is trivial and the groupFI�.K/ is the homology of the two copies of R.K/ which comprise C. Thus,

FI�.unknot/ D H�.C/

D H��R.unknot/qR.unknot/

�D H�.S

2q S2/

D Z2 ˚ Z2:

Noting again Observation 1.1, we can say that FI�.unknot/ is isomorphic totwo copies of the Khovanov homology of the unknot. It is natural to askwhether this isomorphism holds for a larger class of knots:

6

Question 1.2. Is there an isomorphism of abelian groups

FI�.K/ Š Kh.K/˚ Kh.K/

for all alternating knots?

There is evidence for an affirmative answer to this question for the torusknots of type .2; p/, as well as for the (non-alternating) torus knots of type.3; 4/ and .3; 5/. It also seems likely that the answer is in the affirmative forall alternating knots if we use Z=2 coefficients instead of integer coefficients.Already for the .4; 5/ torus knot, however, it is clear from an examination ofR.K/ that the framed instanton homology FI�.K/ has smaller rank than twocopies of the Khovanov homology, so the isomorphism does not extend to allknots.

For a general knot, we expect that Kh.K/ ˚ Kh.K/ is related to FI�.K/through a spectral sequence. There is a similar spectral sequence (though onlywith Z=2 coefficients) relating (reduced) Khovanov homology to the HeegaardFloer homology of the branched double cover [33]. The argument of [33] pro-vides a potential model for a similar argument in the case of our framed instan-ton homology. An important ingredient is to show that there is a long exactsequence relating the framed instanton homologies for K, K0 and K1, whenK0 and K1 are obtained from a knot or link K by making the two differentsmoothings of a single crossing. It seems that a proof of such a long exactsequence can be given using the same ideas that were used in [17] to prove asurgery exact sequence for Seiberg-Witten Floer homology, and the authorshope to return to this and other related issues in a future paper.

Other variations. Forming a connected sum with T 3, as we just did in the def-inition of FI�.Y;K/, is one way to take an arbitrary pair .Y;K/ and modifyit so as to satisfy the non-integrality condition; but of course there are manyother ways. Rather than using T 3, one can use any pair satisfying the non-integrality condition; and an attractive example that – like the 3-torus – carriesisolated flat connections, is the pair .S1 � S2; L/, where L is the 3-componentlink formed from three copies of the S1 factor.

We shall examine this and some other variations of the basic constructionin section 4. Amongst these is a “reduced” version of FI�.K/ that appearsto bear the same relation to reduced Khovanov homology as FI�.K/ does tothe Khovanov homology of K. (For this reduced variant, the homology ofthe unknot is Z.) Another variant arises if, instead of forming a connected

7

sum, we perform 0-surgery on a knot (in S3, for example) and apply the basicconstruction to the core of the surgery torus in the resulting 3-manifold. Thegroup obtained this way is reminiscent of the “longitude Floer homology” ofEftekhary [12]. It is trivial for the “unknot” in any Y (i.e. a knot that boundsa disk), and is Z4 for the trefoil in S3.

In another direction, we can alter the construction of FI�.K/ by dividingthe relevant configuration space by a slightly larger gauge group, and in thisway we obtain a variant of FI�.K/ which we refer to as NFI�.K/ and which is(roughly speaking) half the size of FI�.K/. For this variant, the appropriatemodification of Question 1.2 has only one copy of Kh.K/ on the right-handside. (For example, if K is the unknot, then NFI�.K/ is the ordinary homologyof S2, which coincides with the Khovanov homology.)

Slice-genus bounds. A very interesting aspect of Khovanov homology was dis-covered by Rasmussen [35], who showed how the Khovanov homology of aknot can be used to provide a lower bound for the knot’s slice-genus. Anargument with a very similar structure can be constructed using the framedinstanton homology FI�.K/. The construction begins by replacing Z as thecoefficient group with a certain system of local coefficients � on B.S3; K/. Inthis way we obtain a new group FI�.KI�/ that is finitely-presented moduleover the ring ZŒt�1; t � of finite Laurent series in a variable t . We shall showthat FI�.KI�/ modulo torsion is essentially independent of the knot K: it isalways a free module of rank 2. On the other hand, FI�.KI�/ comes with adescending filtration, and we can define a knot invariant by considering thelevel in this filtration at which the two generators lie. From this knot invariant,we obtain a lower bound for the slice genus.

Although the formal aspects of this argument are modeled on [35], theactual mechanisms behind the proof are the same ones that were first used in[21] and [18].

Monotonicity and other structure groups. The particular conjugacy class cho-sen in (1) is a distinguished one. It might seem, at first, that the definition ofI�.Y;K;P / could be carried out without much change if instead we used anyof the non-central conjugacy classes in SU.2/ represented by the elements

exp 2�i��� 0

0 �

�with � in .0; 1=2/. This is not the case, however, because unless � D 1=4

we cannot establish the necessary finiteness results for the spaces of trajecto-

8

ries in the Morse theory that defines I�.Y;K;P /. The issue is what is usuallycalled “monotonicity” in a similar context in symplectic topology. In general,a Morse theory of Floer type involves a circle-valued Morse function f on aninfinite-dimensional space B whose periods define a map

�f W �1.B/! R:

The Hessian of f may also have spectral flow, defining another map,

sf W �1.B/! Z:

The theory is called monotone if these two are proportional. Varying the eigen-value � varies the periods of the Chern-Simons functional in our theory, andit is only for � D 1=4 that we have monotonicity. In the non-monotone case,one can still define a Morse homology group, but it is necessary to use a localsystem that has a suitable completeness [30].

A related issue is the question of replacing SU.2/ by a general compact Liegroup, say a simple, simply-connected Lie groupG. The choice of � above nowbecomes the choice of an element ˆ of the Lie algebra of G, which will deter-mine the leading term in the singularity of the connections that we use. If wewish to construct a Floer homology theory, then the choice ofˆ is constrainedagain by the monotonicity requirement. It turns out that the monotonicitycondition is equivalent to requiring that the adjoint orbit of ˆ is Kahler-Einstein manifold with Einstein constant 1, when equipped with the Kahlermetric corresponding to the Kirillov-Kostant-Souriau 2-form. We shall de-velop quite a lot of the machinery in the context of a general G, but in the endwe find that it is only for SU.N / that we can satisfy two competing require-ments: the first requirement is monotonicity; the second requirement is thatwe avoid connections with non-trivial stabilizers among the critical points ofthe perturbed Chern-Simons functional. Using SU.N /, we shall construct avariant of FI�.K/ that seems to bear the same relation to the sl.N /Khovanov-Rozansky homology [16] as FI�.K/ does to the original Khovanov homology.

1.3 Discussion

Unlike FI�.K/, the Khovanov homology of a knot is bigraded. If we writePK.q; t/ for the 2-variable Poincare polynomial of Kh.K/, then PK.q;�1/ is(to within a standard factor) the Jones polynomial of K [15]. Relationshipsbetween the Jones polynomial and gauge theory can be traced back to Witten’sreinterpretation of the Jones polynomial as arising from a .2C 1/-dimensional

9

topological quantum field theory [44]. What we are exploring here is in onehigher dimension: a relationship between the Khovanov homology and gaugetheory in 3C 1 dimensions.

Our definition of FI�.Y;K/ seems somewhat unsatisfactory, in that it in-volves a rather unnatural-looking connected sum with T 3. As pointed outabove, one can achieve apparently the same effect by replacing T 3 here withthe pair .S1 � S2; L/ where L is a standard 3-stranded link. The unsatisfac-tory state of affairs is reflected in the fact that we are unable to prove thatthese two choices would lead to isomorphic homology groups. The authorsfeel that there should be a more natural construction, involving the Morsetheory of the Chern-Simons functional on the “framed” configuration spaceQB.Y;K/ D C.Y;K/=G o.Y;K/, where G o � G is the subgroup consisting of

gauge transformations that are 1 at a basepoint y0 2 .Y nK/. The reducedhomology theory would be constructed in a similar manner, but using a base-point k0 on K. A related construction, for homology 3-spheres, appears in analgebraic guise in [8, section 7.3.3].

This idea of using the framed configuration space QB.Y;K/ and dispensingwith the connect-sum with T 3 is attractive: it would enable one to work witha general G without concern for avoiding reducible solutions. However, it can-not be carried through without overcoming obstacles involving bubbles in theinstanton theory: the particular issue is bubbling at the chosen base-point.

Acknowledgments. The development of these ideas was strongly influenced bythe paper of Seidel and Smith [39]. Although gauge theory as such does notappear there, it does not seem to be far below the surface. The first author pre-sented an early version of some of the ideas of the present paper at the Institutefor Advanced Study in June 2005, and learned there from Katrin Wehrheimand Chris Woodward that they were pursuing a very similar program (devel-oping from [42] in the context of Lagrangian intersection Floer homology).Ciprian Manolescu and Chris Woodward have described a similar program,also involving Lagrangian intersections, motivated by the idea of using theframed configuration space. The idea of using a 3-stranded link in S1 � S2 asan alternative to T 3 was suggested to us by Paul Seidel and also appears in thework of Wehrheim and Woodward.

10

2 Instantons with singularities

For the case that the structure group is SU.2/ or PSU.2/, instantons withcodimension-2 singularities were studied in [20, 21] and related papers. Ourpurpose here is to review that material and at the same time to generalize someof the constructions to the case of more general compact groupsG. In the nextsection, we will be considering cylindrical 4-manifolds and Floer homology;but in this section we begin with the closed case. We find it convenient to workfirst with the case that G is simple and simply-connected. Thus our discussionhere applies directly to SU.N / but not to U.N/. Later in this section we willindicate the adjustments necessary to work with other Lie groups, includingU.N/.

For instantons with codimension-2 singularities and arbitrary structuregroup, the formulae for the energy of solutions and the dimension of mod-uli spaces which we examine here are closely related to similar formulae fornon-abelian Bogomoln’yi monopoles. See [28, 29, 43], for example.

2.1 Notation and root systems

For use in the rest of the paper, we set down some of the notation we shall usefor root systems and related matters. Fix G, a compact connected Lie groupthat is both simple and simply connected. We will fix a maximal torus T � Gand denote by t � g its Lie algebra. Inside t is the integer lattice consistingof points x such that exp.2�x/ is the identity. The dual lattice is the latticeof weights: the elements in t� taking integer values on the integer lattice. Wedenote by R � t� the set of roots. We choose a set of positive roots RC � R,so thatR D RC[R�, withR� D �RC. The set of simple roots correspondingto this choice of positive roots will be denoted by �C � RC. We denote by �half the sum of the positive roots, sometimes called the Weyl vector,

� D1

2

Xˇ2RC

ˇ;

and we write � for the highest root.We define the Killing form on g with a minus sign, as

ha; bi D �tr.ad.a/ ad.b//;

so that it is positive definite. The corresponding map g� ! g will be denoted˛ 7! ˛�, and the inverse map is 7! �. If ˛ is a root, we denote by ˛_ the

11

coroot

˛_ D2˛�

h˛; ˛i:

The simple coroots, ˛_ for ˛ 2 �C, form an integral basis for the integer latticein t. The fundamental weights are the elements of the dual basis w˛, ˛ 2 �C

for the lattice of weights. The fundamental Weyl chamber is the cone in t onwhich all the simple roots are non-negative.1 This is the cone spanned by theduals of the fundamental weights, w�˛ 2 t.

The highest root � and the corresponding coroot �_ can be written as pos-itive integer combinations of the simple roots and coroots respectively: thatis,

� DX˛2�C

n˛˛

�_ DX˛2�C

n_˛˛_

for non-negative integers n˛ and n_˛. The numbers

h D 1CX˛2�C

n˛

h_ D 1CX˛2�C

n_˛

(5)

are the Coxeter number and dual Coxeter number respectively. The squaredlength of the highest root is equal to 1=h_:

h�; �i D 1=h_: (6)

(The inner product on g� here is understood to be the dual inner product tothe Killing form.) We also record here the relation

� DX˛2�C

w˛I (7)

this and the previous relation have the corollary

2h�; �i D 1 � 1=h_: (8)

1Note that our convention is that the fundamental Weyl chamber is closed: it is not the locuswhere the simple roots are strictly positive.

12

For each root ˛, there is a preferred homomorphism

j˛ W SU.2/! G

whose derivative maps

dj˛ W

�i 0

0 �i

�7! ˛_:

In the case that G is simply laced, or if ˛ is a long root in the non-simply lacedcase, then the map j˛ is injective and represents a generator of �3.G/. In par-ticular, this applies when ˛ is the highest root � . Under the adjoint action ofj� .SU.2//, the Lie algebra g decomposes as one copy of the adjoint represen-tation of SU.2/, a number of copies of the defining 2-dimensional representa-tion of SU.2/, and a number of copies of the trivial representation. The pair.G; j� .SU.2/// is 4-connected.

2.2 Connections and moduli spaces

Let X be a closed, connected, oriented, Riemannian 4-manifold, and let † �X be a smoothly embedded, compact oriented 2-manifold. We do not assumethat † is connected. We take a principal G-bundle P ! X , where G is asabove. Such a P is classified by an element of �3.G/, and hence by a singlecharacteristic number: following [2], we choose to normalize this characteristicnumber by defining

k D �1

.4h_/p1.g/ŒX�;

where h_ is the dual Coxeter number of G. This normalization is chosen as in[2] so that k takes all values in Z as P ranges through all bundles on X . If thestructure group of P is reduced to the subgroup j� .SU.2//, the k coincideswith the second Chern number of the corresponding SU.2/ bundle.

Fix ˆ 2 t belonging to the fundamental Weyl chamber, and suppose that

�.ˆ/ < 1 (9)

where � is the highest root. This condition is equivalent to saying that

�1 < ˛.ˆ/ < 1

for all roots ˛. This in turn means that an element U 2 g is fixed by theadjoint action of exp.2�ˆ/ if and only if ŒU;ˆ� D 0. In a simply-connectedgroup, the commutant of any element is always connected, and it therefore

13

follows that the subgroup ofG which commutes with exp.2�ˆ/ coincides withthe stabilizer of ˆ under the adjoint action. We call this group Gˆ. We writegˆ � g for its Lie algebra, and we let o stand for the unique G-invariantcomplement, so that

g D gˆ ˚ o: (10)

The set of roots can be decomposed according to the sign of ˛.ˆ/, as

R D RC.ˆ/ [R0.ˆ/ [R�.ˆ/:

Similarly, the set of simple roots decomposes as

�C D �C.ˆ/ [�0.ˆ/;

where �0.ˆ/ are the simple roots that vanish on ˆ. Knowing �0.ˆ/, we canrecover R0.ˆ/ as the set of those positive roots lying in the span of�0.ˆ/. Wecan decompose the complexification o˝C as

o˝C D oC ˚ o� (11)

where oC � g ˝ C is the sum of the weight spaces for all roots ˛ in RC.ˆ/,and o� is the sum of the weight spaces for roots in R�.ˆ/.

Choose a reduction of the structure group ofP j† to the subgroupGˆ � G.Extend this arbitrarily to a tubular neighborhood � � †, so that we have a re-duction of P j� . IfO Š G=Gˆ is the adjoint orbit ofˆ in the Lie algebra g andif OP � gP is the corresponding subbundle of the adjoint bundle, then sucha reduction of structure group to Gˆ � G is determined by giving a section 'of OP over the neighborhood �. We denote the principal Gˆ-bundle resultingfrom this reduction by P' � P j� . There are corresponding reductions of theassociated bundle gP with fiber g and its complexification, which we write as

gP j� D g' ˚ o' (12)

and.gP ˝C/j� D .g' ˝C/˚ .o' ˝C/

D .g' ˝C/˚ oC' ˚ o�' :

We identify � diffeomorphically with the disk bundle of the oriented nor-mal bundle to †, we let i� be a connection 1-form on the circle bundle @�, andwe extend � by pull-back to the deleted tubular neighborhood �n†. Thus �is a 1-form that coincides with d� in polar coordinates .r; �/ on each normaldisk under the chosen diffeomorphism.

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The data ' and � together allow us to define the model for our singularconnections. We choose a smooth G-connection A0 on P . We take ˇ.r/ tobe a cutoff-function equal to 1 in a neighborhood of † and supported in theneighborhood �. Then we define

A' D A0 C ˇ.r/' ˝ � (13)

as a connection in P over Xn†. (The form ˇ.r/'� has been extended by zeroto all ofXn†.) The holonomy ofA' around a loop r D � in a normal disk to†(oriented with the standard � coordinate increasing) defines an automorphismof P over �n† which is asymptotically equal to

exp.�2�'/ (14)

when � is small.Following [20], we fix a p > 2 and consider a space of connections on

P jXn† defined as

Cp.X;†;P; '/ D fA' C a j a;rA'a 2 Lp.Xn†/ g: (15)

As in [20, Section 3], the definition of this space of connections can be refor-mulated to make clear that it depends only on the reduction of structure groupdefined by ', and does not otherwise depend on ˆ. To do this, extend theradial distance function r as a positive function on Xn† and define a Banachspace W p

k.X/ for k � 1 by taking the completion of the compactly supported

smooth functions on Xn† with respect to norm

kf kW p

kD

1rk f p

C

1

rk�1rf

p

C � � � C

rkf p:

(For k D 0, we just define W p

kto be Lp.) The essential point then is that the

condition on aj� that arises from the definition (15) can be equivalently written(using the decomposition (12)) as

aj� 2 Lp1 .�Ig'/˚W

p1 .�I o'/:

This shows that the space Cp.X;†;P; '/ depends only on the decompositionof gP as g' ˚ o' . It is important here that the condition (9) is satisfied: thiscondition ensures that the eigenvalues of the bundle automorphism (14) actingon o' are all different from 1, for these eigenvalues are exp.˙2�i˛.ˆ// as ˛runs throughRC.ˆ/. This space of connections is acted on by the gauge group

Gp.X;†;P; '/ D fg 2 Aut.P jXn†/ j rA'g;r2A'g 2 L

p.Xn†/ g:

15

The fact that this is a Banach Lie group acting smoothly on Cp.X;†;P; '/ isa consequence of multiplication theorems, such as the continuity of the multi-plications W p

2 �Wp1 ! W

p1 , just as in [20].

The center of the gauge group Gp D Gp.X;†;P; '/ is canonically iso-morphic to the center Z.G/ of G, a finite group. This subgroup Z.Gp/ actstrivially on the Cp.X;†;P; '/, so the group that acts effectively is the quotientGp=Z.Gp/. Some connections A will have larger stabilizer; but there is an im-portant distinction here that is not present in the most familiar case, whenG D SU.N /. In the case of SU.N / if the stabilizer of A is larger than Z.Gp/,then the stabilizer has positive dimension, but for otherG the stabilizer may bea finite group strictly larger than G. To understand this point, recall that thestabilizer of a connection A in the gauge group is isomorphic to the central-izer CG.S/ where S is the set of holonomies around all loops based at somechosen basepoint. So the question of which stabilizers occur is equivalent tothe question of which subgroups of G arise as CG.H/ for some H � G, whichwe may as well take to be a closed subgroup. In the case of SU.N /, the onlyfinite group that arises this way is the center. But for other simple Lie groupsG, there may be a semi-simple subgroup H � G of the same rank as G; andin this case the centralizer CG.H/ is isomorphic to the center ofH , which maybe strictly larger than the center of G. Examples of this phenomenon includethe case where G D Spin.2nC1/ andH is the subgroup Spin.2n/. In this caseCG.H/ contains Z.G/ as a subgroup of index 2. Another case is G D G2 andH D SU.3/: in this case CG.H/ has order 3 while the center of G is trivial.

We reserve the word reducible for connections A whose stabilizer has posi-tive dimension:

Definition 2.1. We will say that a connection A is irreducible if its stabilizer inthe gauge group is finite.

The homotopy type of Gp.X;†;P; '/ coincides with that of the group ofall smooth automorphisms ofP which respect the reduction of structure groupalong†. The bundle P is classified its characteristic number k, and the section' is determined up to homotopy by the induced map on cohomology,

'� W H 2.OP /! H 2.†/: (16)

Because the restriction of P to† is trivial, and because the choice of trivializa-tion is unique up to homotopy, we can also think of the reduction of structuregroup along† as being determined simply by specifying the isomorphism classof the principal Gˆ-bundle P' ! †. In this way, when † is connected, the

16

classification is by �1.Gˆ/. The inclusion T ! Gˆ induces a surjection on �1,so we can lift to an element of �1.T /, which we can reinterpret as an integerlattice point � in t. Let Z.Gˆ/ � T be the center of Gˆ, let z.Gˆ/ be its Liealgebra, and let … be the orthogonal projection

… W t! z.Gˆ/:

We can describe z.Gˆ/ as

z.Gˆ/ D\

˛2S0.ˆ/

ker˛

D spanfw�ˇj ˇ 2 SC.ˆ/ g:

The projection ….�/ may not be an integer lattice point, but the image under… of the integer lattice in t is isomorphic to �1.Gˆ/, and the reduction ofstructure group is determined up to homotopy by the element

….�/ 2 z.Gˆ/:

We give the image of the integer lattice under … a name:

Definition 2.2. We write L.Gˆ/ � z.Gˆ/ for the image under … of the integerlattice in t. Thus L.Gˆ/x is isomorphic both toH2.OIZ/ and to �1.Gˆ/, andclassifies the possible reductions of structure group of P ! † to the subgroupGˆ, in the case that † is connected. If the reduction of structure group deter-mined by ' is classified by the lattice element l 2 L.Gˆ/, we refer to l as themonopole charge. If † has more than one component, we define the monopolecharge by summing over the components of †.

We now wish to define a moduli space of anti-self-dual connections as

M.X;†;P; '/ D fA 2 Cp j FCA D 0 gı

Gp:

As shown in [20], there is a Kuranishi model for the neighborhood of a con-nection ŒA� in M.X;†;P; '/ described by a Fredholm complex, as long as pis chosen sufficiently close to 2. Specifically, if x denotes the smallest of all thereal numbers ˛.ˆ/ and 1� ˛.ˆ/ as ˛ runs through RC.ˆ/, then p needs to bein the range

2 < p < 2C �.x/ (17)

where � is a continuous function which is positive for x > 0 but has �.0/ D 0.We suppose henceforth that p is in this range. The Kuranishi theory then tellsus, in particular, that if A is irreducible and the operator

dCA W Lp1;A.Xn†IgP ˝ƒ

1/! Lp.Xn†IgP ˝ƒC/

17

is surjective, then a neighborhood of ŒA� inM.X;†;P; '/ is a smooth manifold(or orbifold if the finite stabilizer is larger than Z.G/), and its dimension isequal to minus the index of the Fredholm complex

Lp2;A.Xn†IgP ˝ƒ

0/dA! L

p1;A.Xn†IgP ˝ƒ

1/dC

A! Lp.Xn†IgP ˝ƒ

C/:

No essential change is needed to carry over the proofs from [20]. The non-linear aspects come down to the multiplication theorems for the W p

kspaces,

while the Fredholm theory for the linear operators reduces in the end to thecase of a line bundle with the weighted norms.

We refer to minus the index of the above complex as the formal dimensionof the moduli space M.X;†;P; '/. We can write the formula for the formaldimension as

.4h_/k C 2˝c1.o

�' /; Œ†�

˛C.dimO/

2�.†/ � .dimG/.bC � b1 C 1/: (18)

Here dimO denotes the dimension of O as a real manifold: an even number,because O is also a complex manifold. The proof of this formula can be givenfollowing [20] by using excision to reduce to the simple case that the reductionof structure group can extended to all of X . In this way, the calculation caneventually be reduced to calculating the index of a Fredholm complex of thesame type as above, but with gP replaced by a complex line-bundle o� on X ,equipped with a singular connection d� of the form

r C iˇ.r/��

with� 2 .0; 1/ (cf. (13)). The index calculation in the case of such a line bundleis given in [20].

We can express the characteristic class c1.o�' / that appears in the formula(18) in slightly different language. As was just mentioned, the manifold Ois naturally a complex manifold. To define the standard complex structure, weidentify the tangent space toO atˆwith o, and we give o a complex structure Jby identifying it with o� � o˝C using the linear projection. This gives a Gˆ-invariant complex structure on TˆO which can be extended to an integrablecomplex structure on all of O using the action of G. The bundle OP ! X isnow a bundle of complex manifolds, and we use c1.OP =X/ to denote the firstChern class of its vertical tangent bundle. Then we can rewrite c1.o�' / as

c1.o�' / D '

�.c1.OP =X//

18

inH 2.†/. Using again the canonical trivialization of P j† up to homotopy, wecan also think of ' as simply a map†! O up to homotopy, and we can thinkof the characteristic class as '�.c1.O//.

We can summarize the situation with a lemma:

Lemma 2.3. Let the reduction of structure group of P ! † to the subgroupGˆ have monopole charge l . Then the formula (18) for formal dimension of themoduli space can be rewritten as

4h_k C 4�.l/C.dimO/

2�.†/ � .dimG/.bC � b1 C 1/; (19)

where � is, as above, the Weyl vector.

Proof. The difference between this expression and the previous formula (18) isthe replacement of the term involving 2c1.o�' /Œ†� by the term involving 4�.l/.To see that these are equal, we may reduce the structure group of theGˆ bundleover † to the torus T , and again write � for the vector in the integer lattice ofT that classifies this T -bundle. The bundle o�' now decomposes as a direct sumaccording to the positive roots in RC.ˆ/:

o�' DM

˛2RC.ˆ/

o�˛ :

We havec1.o

�˛ /Œ†� D ˛.�/I

so,c1.o

�' /Œ†� D

X˛2RC.ˆ/

˛.�/

D

X˛2RC.ˆ/

h˛�; �i:

For each simple root ˇ in S0.ˆ/, the Weyl group reflection �ˇ permutes thevectors

f˛� j ˛ 2 RC.ˆ/ g:

It follows that when we writeX˛2RC

˛� DX

˛2RC.ˆ/

˛� CX

˛2R0.ˆ/\RC

˛�;

the first term on the right is in the kernel of ˇ for all ˇ in S0.ˆ/; i.e. the firstterm belongs to z.Gˆ/. The second term on the right belongs to the orthogonal

19

complement of z.Gˆ/, because it is in the span of the elements ˛_ as ˛ runsthrough S0.ˆ/. Recalling the definition of the Weyl vector �, we thereforededuce X

˛2RC.ˆ/

˛� D 2…��:

Thusc1.o

�' /Œ†� D

X˛2RC.ˆ/

h˛�; �i

D 2h…��; �i

D 2h�;…�i

D 2�.l/

as desired, because l D …�.

2.3 Energy and monotonicity

Along with the formula (19) for the dimension of the moduli space, the otherimportant quantity is the energy of a solution A in Cp.X;†;P; '/ to the equa-tions FCA D 0, which we define as

E D 2

ZXn†

jFAj2 dvol

D 2

ZXn†

�tr.ad.�FA/ ^ ad.FA//:(20)

(Note that the norm on gP in the first line is again defined using the Killingform, �tr.ad.a/ ad.b//.) The reason for the factor of two is to fit with our useof the path energy in the context of Floer homology later.

This quantity depends only on P and ', and can be calculated in termsof the instanton number k and the monopole charges l . To do this, we againsuppose that the structure group of P' ! † is reduced to the torus T , and wedecompose the bundle oC' again as (11). The formula for E as a function of Pand ' can then be written, following the argument of [20], as

E.X;†;P; '/ D 32�2�h_k C

Xˇ2RC

ˇ.ˆ/c1.o�ˇ /Œ†� �

1

2

Xˇ2RC

ˇ.ˆ/2.† �†/�

D 32�2�h_k C

Xˇ2RC

ˇ.ˆ/ˇ.�/ �1

2

Xˇ2RC

ˇ.ˆ/2.† �†/�

(21)

20

where � is again the integer vector in t classifying the T -bundle to which P'has been reduced. The two sums involving the positive roots in this formulaeach be interpreted as half the Killing form, which leads to the more compactformula

E D 8�2�4h_k C 2hˆ; �i � hˆ;ˆi.† �†/

�:

Finally, using the fact that ˆ belongs to z.Gˆ/, we can replace � by its projec-tion l and write

E D 8�2�4h_k C 2hˆ; li � hˆ;ˆi.† �†/

�: (22)

An important comparison to be made is between the linear terms in k andl in the formula for E and the linear terms in the same variables in the formula(19) for the formal dimension of the moduli space. In the dimension formula,these linear terms are

4h_k C 4�.l/ (23)

while in the formula for E they are

8�2�4h_k C 2hˆ; li

�: (24)

Definition 2.4. We shall say that the choice ofˆ is monotone if the linear forms(24) and (23) in the variables k and l are proportional.

Proposition 2.5. Let ˆ0 be any element in the fundamental Weyl chamber. Thenthere exists a unique ˆ in the same Weyl chamber such that the stabilizers of ˆ0and ˆ coincide, and such that the monotonicity condition holds. Furthermore,this ˆ satisfies the constraint (9).

Proof. We are seeking a ˆ with Gˆ D Gˆ0 . The Lie algebras of these groupstherefore have the same center, in which both ˆ and ˆ0 lie, and we can write… for the projection of t onto the center. From the formulae, we see that themonotonicity condition requires that hˆ; li D 2�.l/ for all l in the image of….This condition is satisfied only by the element

ˆ D 2….��/: (25)

We can rewrite this asˆ D

X˛2RC.ˆ0/

˛�: (26)

21

It remains only to verify the bound �.ˆ/ < 1. But we have

�.ˆ/ <X˛2RC

�.˛�/

D 2h�; �i

D 1 � 1=h_

< 1

by (8), as desired.

2.4 Geometric interpretation of the monotonicity condition

We can reinterpret these formulae in terms of cohomology classes on O. As ageneral reference for the following material, we cite [3, Chapter 8]. In the firstline of (21), we see the characteristic classX

ˇ2RC

ˇ.ˆ/c1.o�ˇ /: (27)

The decomposition of o�' as the sum of o�ˇ

reflects our reduction of the struc-ture group to T . A more invariant way to decompose this bundle is as follows.We writeEC � z.Gˆ/

� for the set of non-zero linear forms on z.Gˆ/ arising asˇjz.Gˆ/ for ˇ in RC. The elements of EC are weights for the action of Z.Gˆ/on o, and we have a corresponding decomposition of the vector bundle o�' intoweight spaces,

o�' DM 2EC

o�' . /:

The characteristic class (27) can be written more invariantly asX 2EC

.ˆ/c1.o�' . //: (28)

The tangent bundle TO has a decomposition of the same form, as a complexvector bundle,

TO DM 2EC

TO. /:

Using the canonical trivialization of P ! † up to homotopy, interpret ' againas a map

' W †! O;

22

and then rewrite the characteristic class (28) as '�.�ˆ/, where

�ˆ DX 2EC

.ˆ/c1.TO. // 2 H2.OIR/: (29)

In this way, we can rewrite the linear form (24) in k and l as

32�2�h_k C

˝'��ˆ; Œ†�

˛�(30)

Using this last expression, we see that the monotone condition simply re-quires that

c1.o�/ D 2�ˆ (31)

in H 2.OIR/. This equality has a geometric interpretation in terms of the ge-ometry of the orbit O of ˆ in g. Recall that we have equipped O with a com-plex structure J , so that its complex tangent bundle is isomorphic to o�. Thereis also the Kirillov-Kostant-Souriau 2-form on O, which is the G-invariantform !ˆ characterized by the condition that at ˆ 2 O it is given by

!ˆ.ŒU;ˆ�; ŒV;ˆ�/ D˝ˆ; ŒU; V �

˛where the angle brackets denote the Killing form. Together, J and !ˆ makeO into a homogeneous Kahler manifold. The cohomology class Œ!ˆ� of theKirillov-Kostant-Souriau form is 4��ˆ, so the monotonicity condition canbe expressed as

Œ!ˆ� D 2�c1.O/:

The fact that the Kahler class and the first Chern class are proportional means,in particular, that .O; !ˆ/ is a monotone symplectic manifold, in the usualsense of symplectic topology. In the homogeneous case, this proportional-ity (with a specified constant) between the classes Œ!ˆ� and c1.O/ on O im-plies a corresponding relation between their natural geometric representatives,namely !ˆ itself and the Ricci form. That is to say, our monotonicity condi-tion is equivalent to the equality

gˆ D Ricci.gˆ/

for the Kahler metric gˆ corresponding to !ˆ. Thus in the monotone case, Ois a Kahler-Einstein manifold with Einstein constant 1.

23

2.5 The case of the special unitary group

We now look at the case of the special unitary group SU.N /. We continueto suppose that X � † is a pair consisting of a 4-manifold and an embeddedsurface, as in the previous subsections. Let P ! X be a given principal SU.N /bundle. An element ˆ in the standard fundamental Weyl chamber in the Liealgebra su.N / has the form the form

ˆ D i diag.�1; : : : ; �1; �2; : : : ; �2; : : : ; �m; : : : ; �m/

where �1 > �2 > � � � > �m. Let the multiplicities of the eigenspaces beN1; : : : ; Nm, so that N D

PNs. We will suppose that

�1 � �N < 1; (32)

this being the analog of the condition (9). Let O � su.N / be the orbit of ˆunder the adjoint action. Choose a section ' of the associated bundle OP j†,so defining a reduction of the structure group of P j† to the subgroup

S.U.N1/ � � � � � U.Nm//:

Let Ps ! † be the principal U.N C s/-bundle arising from the s’th factor inthis reduction and define

ls D �c1.Ps/Œ†�:

Because we have a special unitary bundle, we haveXNs�s D 0X

ls D 0:

The integers ls are equivalent data to the monopole charge l in L.Gˆ/ as de-fined in Definition 2.2 for the general case: more precisely, the relationshipis

l D i diag.l1=N1; : : : ; l1=N1; : : : ; lm=Nm; : : : ; lm=Nm/:

The Killing form is 2N times the standard trace norm on su.N /, so we havefor example

hˆ; li D 2N

mXsD1

�sls

The dual Coxeter number is N and the energy formula becomes:

E.X;†;P; '/ D 32�2N�k C

mXsD1

�sls �12

� mXsD1

�2sNs�† �†

�:

24

If Es is the vector bundle associated to Ps by the standard representation,then the bundle denoted o�' in the previous sections (the pull-back by ' of thevertical tangent bundle of OP , equipped with its preferred complex structure)can be written as

o�' DMs<t

E�s ˝Et ;

and the first Chern class of this bundle evaluates on † as

c1.o�' /Œ†� D

mXsD1

mXtD1

sign.t � s/Nt ls:

The dimension formula becomes

4NkC2Xs;t

sign.t � s/Nt lsC�Xs<t

NsNt

��.†/� .N 2

�1/.bC�b1C1/: (33)

where k is the second Chern number of P . Thus the monotone conditionsimply requires that

�s D1

2N

mXtD1

sign.t � s/Nt (34)

for all s. Notice that the ˆ whose eigenvalues are given by the formula (34)with multiplicities Ns is already traceless, and satisfies the requirement (32),which we can take as confirming Proposition 2.5 in the case of SU.N /.

Examples. (i) The simplest example occurs when P is an SU.2/ bundle andthe reduction of structure group is to U.1/. In this case, we can write

ˆ D i

�� 0

0 ��

�with � 2 .0; 1=2/, and the holonomy of A' along small loops linking † isasymptotically

exp 2�i��� 0

0 �

�:

We can write .l1; l2/ as .l;�l/ and formula for the index becomes

8k C 4l C �.†/ � 3.bC � b1 C 1/:

The action is given by

64�2.k C 2�l � �2† �†/:

25

These are the formulae from [20]. The monotone condition requires that � D1=4, and in this case the asymptotic holonomy on small loops is�

�i 0

0 i

�:

(ii) The next simplest case is that of SU.N / with two eigenspaces, of di-mensions N1 D 1 and N2 D N � 1. In this case, O becomes CPN�1. We canagain write .l1; l2/ as .l;�l/ and we can write

�1 D �

�2 D ��=.N � 1/

with � in the interval .0; 1/. The formula for the dimension is

4Nk C 2N l C .N � 1/�.†/ � .N 2� 1/.bC � b1 C 1/

and the energy is given by

32�2N�k C

N

N � 1

��l � 1

2�2† �†

��:

The monotone condition requires � D .N � 1/=.2N /, so that

�1 D .N � 1/=.2N /

�2 D �1=.2N /:

The asymptotic holonomy on small loops is

� diag.�1; 1; : : : ; 1/

where � D e�i=N is a .2N /’th root of unity.

2.6 Avoiding reducible solutions

We return to the case of a generalG (still simple and simply connected). Recallthat a connection is reducible if its stabilizer in Gp.X;†;P; '/ has positivedimension. This is equivalent to saying that there is a non-zero section ofthe bundle gP on Xn† that is parallel for the connection A.

Let us ask under what conditions this can occur for an ŒA� belonging to themoduli space M.X;†;P; '/. For simplicity, we suppose for the moment that† is connected. If is parallel, then it determines a single orbit in g; we take‰ 2 g to be a representative. The structure group of the bundle then reduces

26

to the subgroup G‰, the stabilizer of ‰, which is a connected proper subgroupof G. Because ˆ and ‰ must commute, we may suppose they both belong tothe Lie algebra t of the maximal torus. As before, we shall suppose ˆ belongsto the fundamental Weyl chamber.

The center of G‰ contains a torus of dimension at least one, because ‰itself lies in the Lie algebra of the center. So there is a non-trivial character,

s W G‰ ! U.1/:

Because G‰ contains T , this character corresponds to a weight for this maxi-mal torus: there is an element w 2 t� in the lattice of weights such that

s.exp.2�x// D exp.2�w.x// (35)

for x in t. The fundamental group of T maps onto that of G‰, so we mayassume that w is a primitive weight (i.e. is not a non-trivial multiple of anotherinteger weight).

In addition to taking w to be primitive, we can further narrow down thepossibilities as follows. Suppose first that ‰ lies (like ˆ) in the fundamentalWeyl chamber. In the complex group Gc , there are r D rank.G/ differentmaximal parabolic subgroups which contain the standard Borel subgroup cor-responding to our choice of positive roots. These maximal parabolics are in-dexed by the set of simple roots �C; we let G.˛/, for ˛ 2 �C, denote theintersection of these groups with the compact group G. Each group G.˛/ � Ghas 1-dimensional center, and the weight w corresponding to a primitive char-acter of G.˛/ is the fundamental weight w˛. The group G‰ lies inside one ofthe G.˛/, so the same fundamental weight w˛ defines a character of G‰. If‰ does not lie in the fundamental Weyl chamber, then we need to apply anelement of the Weyl group W ; so in general, we can always take w to have theform

w D w˛ B �

where � 2 W and w˛ is one of the fundamental weights.Applying the character w to the singular connection A, we obtain a sin-

gular U.1/ connection on Xn†, carried by the bundle obtained by applyings to the principal G‰-bundle P ! X . That is, we have a U.1/ connectiondiffering by terms of regularity Lp1;A from the singular model connection

r C iˇ.r/w.ˆ/�:

Here it is important that † is connected: what we have really done, is pickeda base-point near †, and used the values of � and at that base-point to

27

determine T and s. When we apply the Chern-Weil to obtain an expressionfor c1 of the line bundle s.P /, we obtain an additional contribution from thesingularity, equal to the Poincare dual of

2�w.ˆ/Œ†�:

More precisely, if F denotes the curvature of this U.1/ connection on Xn† (asan Lp form, extended by zero to all of X ), then

i

2�ŒF � D c1.s.P // � w.ˆ/P:D:Œ†�:

If† has more than one component, say† D †1[ : : : †r , then the only changeis that we will see a different element of the Weyl group for each component of†: so if we have a reducible solution then there will be a fundamental weightw˛ and elements �1; : : : ; �r in W such that the curvature F of the correspond-ing U.1/ connection satisfies

i

2�ŒF � D c1.s.P // �

rXjD1

.w˛ B �j /.ˆ/P:D:Œ†j �: (36)

Because the connection A is anti-self-dual, the 2-form F is also anti-self-dual, and is therefore L2-orthogonal to every closed, self-dual form h on X .By the usual argument [6], we deduce:

Proposition 2.6. Suppose that bC.X/ � 1, and let the components of † be†1; : : : ; †r . Suppose that for every fundamental weight w˛ and every choiceof elements �1; : : : ; �r in the Weyl group, the real cohomology class

rXjD1

.w˛ B �j /.ˆ/P:D:Œ†j � (37)

is not integral. Then for generic choice of Riemannian metric on X , there are noreducible solutions in the moduli space M.X;†;P; '/.

Examples. As an illustration, in the case G D SU.N /, if † is connected andŒ†� is primitive, then for (37) to be an integral class means that the sum ofsome proper subset of the eigenvalues of ˆ (listed with repetitions) is equal toan integer multiple of i . If there are only two distinct eigenvalues i�1 and i�2of multiplicities N1 and N2, and if we are in the monotone case, so that

�1 D N2=.2N /

�2 D �N1=.2N /;

28

then this integrality means that

.2N /ˇ.aN2 � bN1/

for some non-negative with a � N1, b � N2 and 0 < aC b < N . This cannothappen if N1 and N2 are coprime.

Another family of examples satisfying the monotone condition occurswhen ˆ D 2�� (the case where the reduction of structure group is to the maxi-mal torus, T � G) and the group G is small. For example, if ˆ D 2�� and G iseither the group G2 or the simply-connected group of type B2, B3 or B4, then(37) is never an integer. Thus we have:

Corollary 2.7. Suppose that bC.X/ is positive and that † is connected and liesin a primitive homology class. Suppose ˆ is chosen to satisfy the monotone con-dition and that we are in one of the following cases:

(i) G is the group SU.N /, andˆ has two distinct eigenvalues whose multiplic-ities are coprime.

(ii) G is the group G2 and ˆ is the regular element 2��.

(iii) G is the group Spin.5/, Spin.7/ or Spin.9/ and ˆ is 2��.

Then for generic choice of Riemannian metric on X , there are no reducible solu-tions in the moduli space M.X;†;P; '/.

Proof. We illustrate the calculation in the case of Spin.9/. The B4 root systemis the following collection of integer vectors in R4: the vectors ˙ei ˙ ej fori ¤ j , and the vectors˙ei . The simple roots are

˛i D ei � eiC1; .i D 1; 2; 3/;

˛4 D e4

and the fundamental weights are

w1 D .1; 0; 0; 0/

w2 D .1; 1; 0; 0/

w3 D .1; 1; 1; 0/

w4 D .1=2; 1=2; 1=2; 1=2/:

29

The orbit of these under the Weyl group consists of all vectors of the form

˙ei

˙ei ˙ ej

˙ei ˙ ej ˙ ek

.1=2/.˙e1 ˙ e2 ˙ e3 ˙ e4/

(38)

with i; j; k distinct. We identify t with its dual using the Euclidean inner prod-uct on R4, so that the coroots are ˛_i D ˛i for i D 1; 2; 3 and ˛_4 D 2˛4; andwe note that the Killing form on t is 14 times the Euclidean inner product, be-cause the dual Coxeter number is 7. Then we calculate the sum of the positiveroots to obtain

2� D .7; 5; 3; 1/

and we deduce2�� D .1=14/.7; 5; 3; 1/:

It is straightforward to see that 2�� does not have integer pairing with any ofthe vectors in (38), with respect to the Euclidean inner product on R4.

The examples in this corollary are not meant to be exhaustive: the authorshave not attempted a complete classification of the monotone cases. Note thatwe cannot extend the above proof for Spin.9/ to the case of Spin.11/ (the B5Dynkin diagram) because the vector 2�� is then

2��.1=18/.9; 7; 5; 3; 1/

which has inner product 0 in R5 with the vector .1; 0;�1;�1;�1/, which be-longs to the Weyl orbit of the fundamental weight .1; 1; 1; 1; 0/.

Remark. We recall again from section 2.2 that in cases other than SU.N /, it ispossible for a connection to be irreducible and yet have finite stabilizer strictlylarger than the center of the group G. The examples of this phenomenon thatwere illustrated previously include the case thatG is Spin.2nC1/ as well as thecase of G2. Note that these examples include the examples mentioned in part(iii) of Corollary 2.7; so although we can avoid stabilizers of positive dimensionin those cases, we will still be left with finite stabilizers larger than the center.

The examples in Corollary 2.7 are cases where † is connected. The nextcorollary exhibits an interesting case where Proposition 2.6 can be applied to adisconnected †:

30

Corollary 2.8. Suppose bC.X/ is positive. Let G D SU.N / and suppose † hasN C 1 components, all belonging to the same primitive homology class. Let ˆ bethe element

ˆ D

�i

2N

�diag

�.N � 1/;�1; : : : ;�1

�so that the monotone condition holds. Then for a generic choice of Riemannianmetric on X , the moduli space M.X;†;P; '/ contains no reducible solutions.

Proof. From Proposition 2.6, we see that we must check that the rational num-ber

N�1XjD1

.w˛ B �j /.ˆ/

is never an integer. As a function on the maximal torus, the fundamentalweight ˛ can be taken to be the sum of the first k eigenvalues for some k with1 � k � N � 1, and so

.w˛ B �j /.ˆ/ D

(�k=.2N /; or

.N � k/=.2N /;(39)

according to which Weyl group element �j is involved. The above sum is there-fore

.s=2/ � k.N � 1/=.2N /

where k depends on the choice of ˛, and s is the number of components forwhich the second case of (39) occurs. This quantity differs from an element of.1=2/Z by k=.2N /, so it cannot be an integer.

2.7 Bubbles

Uhlenbeck’s compactness theorem for instanton moduli spaces on a closed 4-manifoldX carries over to the case of instantons with codimension-2 singular-ities along a surface† � X . In the case of SU.2/, the proof can again be foundin [20]. The proof carries over without substantial change to the case of a gen-eral group. We shall state here the version appropriate for a simply-connectedsimple Lie group G and a moduli space M.X;†;P; '/ of anti-self-dual con-nections with reduction along †.

Proposition 2.9. Let ŒAn� be a sequence of gauge-equivalence classes of connec-tions in the moduli spaceM.X;†;P; '/. Then, after replacing this sequence by asubsequence, we can find a bundle P 0 ! X , a section '0 of the bundle OP 0 ! †

31

defining a reduction of structure group to the same subgroup Gˆ � G, an ele-ment ŒA� in M.X;†;P 0; '0/ and a finite set of point x � X with the followingproperties.

(i) There is a sequence of isomorphisms of bundles gn W P 0jXnx ! P suchthat g�n.'/ D '

0j†nx and such that

g�n.An/! AjXnx

on compact subsets of Xnx.

(ii) In the sense of measures on X , the energy densities 2jFAn j2 converge to

2jFAj2C

Xx2x

�xıx

where ıx is the delta-mass at x and �x are positive real numbers.

(iii) For each x 2 x, we can find an integer kx and an lx in the lattice L.Gˆ/ �z.Gˆ/ such that

�x D 8�2.4h_kx C 2hˆ; lxi/

If x 62 †, we can take lx D 0 here. Furthermore, if .k; l/ and .k0; l 0/are the instanton numbers and monopole charges for .P; '/ and .P 0; '0/respectively, then we can arrange that

k D k0 CXx2x

kx

l D l 0 CXx2x

lx

(iv) For each such pair .kx; lx/ with x 2 † we can find an expression for theseas finite sums,

kx D kx;1 C � � � C kx;m

lx D lx;1 C � � � C lx;m

and solutions ŒAx;i � in moduli spaces M.S4; S2; Px;i ; 'x;i / for the roundmetric on .S4; S2/, where Px;i is the G-bundle on S4 with k.Px;i / D kx;iand 'x;i is the reduction of structure group along S2 classifies by the ele-ment lx;i / in L.Gˆ/.

32

The content of the last three parts of the proposition is that the energy �xthat is “lost” at each of the point x in x is accounted for by the energy of acollection of solutions on .S4; S2/ that have bubbled off. (The expression

8�2.4h_k C 2hˆ; li/ (40)

is the formula for the energy in the case of .S4; S2/.) In general, if no multipleof ˆ is an integer point, then the set of values realized by this function of kand l is dense in the real line; and while the proof of Uhlenbeck’s theoremdoes provide us with a constant � and a guarantee that �x � � in all cases,we need better information than this to make use of the compactness theoremin applications. For example, the statement of the result as given does notguarantee that the formal dimension of M.X;†;P; '/ is not larger than thatof M.X;†;P 0; '0/.

The essential matter is to know which pairs .k; l/ in Z�L.Gˆ/ are realizedby solutions on .S4; S2/.

Proposition 2.10. Letˆ as usual lie in the fundamental Weyl chamber and satisfythe necessary constraint �.ˆ/ < 1. Then for any solution ŒA� on .S4; S2/ withthe round metric, the corresponding topological invariants .k; l/ in Z � L.Gˆ/must satisfy the inequalities

k � 0 (41)

andn_˛k C w˛.l/ � 0 (42)

for all simple roots ˛.

Remark. We will see in the course of the proof that the above inequalities areequivalent to a smaller set, namely the set consisting of the inequality (41)together with the inequalities (42) taken only for those ˛ belonging to the setof simple roots ˛ in SC.ˆ/ (the simple roots which are positive on ˆ).

Before proving the proposition, we note an important corollary for the for-mal dimensions of the non-empty moduli spaces on .S4; S2/. The dimensionformula in this case can be written

4h_l C 4h��; li � dimGˆ:

We can interpret the first two terms

4h_l C 4h��; li (43)

33

as the dimension of a framed moduli space, as follows. The gauge groupGp.X;†;P; '/ consists of continuous automorphisms of the bundle P ! S4

which preserve the section ' of the adjoint bundle; so if we pick a points 2 S2 � S4 then there is a closed subgroup G

p1 consisting of elements with

g.s/ D 1. The formula (43) can be interpreted as the formal dimension of amoduli space QM.S4; S2/ where we divide the space of anti-self-dual singularconnections by the smaller group G

p1 instead of the full gauge group. We then

have:

Corollary 2.11. For any non-empty moduli space on .S4; S2/ with the roundmetric, the corresponding instanton number k and monopole charge are eitherboth zero (in which case the moduli space contains only the flat connection) orsatisfy

4h_k C 4h��; li � 4:

Proof of Corollary 2.11. We recall the relation (5) and the fact that � is the sumof the w˛ (taken over all simple roots ˛/. Using these, we see that the sum ofall the inequalities in Proposition 2.10 gives us

h_k C h��; li � 0:

To refine this a little, let us break up the sum into two parts according towhether ˛ lies in SC.ˆ/ or S�.ˆ/: we obtain

4h_k C 4h��; li D 4k C 4X

˛2SC.ˆ/

�h_k C hw�˛; li

�C 4

Xˇ2S0.ˆ/

�h_k C hw

�

ˇ; li�

� 4k C 4X

˛2SC.ˆ/

�h_k C hw�˛; li

�:

As well as being non-negative by the proposition, the terms under the finalsummation sign are all integers: this is because l is the projection in z.Gˆ/ ofan integer vector � 2 t and � � l lies in the kernel of w˛ for ˛ in SC.ˆ/. (Thesimilar terms involving the ˇ in S0.ˆ/ need not be integers.)

This shows that 4h_k C 4h��; li is at least 4 unless k is zero and w˛.l/ iszero for all ˛ in SC.ˆ/. These are independent linear conditions which implythat k and l are both zero.

Corollary 2.12. In the situation of Proposition 2.9, if the set of bubble-points xis non-empty, then the formal dimension of the moduli space M.X;†;P 0; '0/ issmaller than the dimension of M.X;†;P; '/, and the difference is at least 4.

34

Proof of Corollary 2.12. The difference in the dimensions is equal to a sumXx

XiD1

�4h_kx;i C 4h�

�; lx;i i�

and each term is at least four by the previous corollary and the condition inpart (iv) of Proposition 2.9.

Proof of Proposition 2.10. The proof rests on a theorem of Munari [27] whichprovides a correspondence between moduli spaces of singular instantons in.S4; S2/ and certain complex-analytic moduli spaces for holomorphic data onCP2. To state Munari’s theorem, fixˆ as usual, letP ! S4 be aG-bundle and' a section of OP ! S2 defining a reduction of structure group. Pick a points 2 S2 and let QM.S4; S2; P; '/ be the corresponding framed moduli space. Let� W CP2 ! S4 be a map which collapses the line at infinity `1 � CP2 to thepoint s and which maps another complex line † to S2 � S4. Write s1 2 CP2

for the point where † and `1 meet. Then we have:

Theorem 2.13 ([27]; see also [4]). There is a bijection between the moduli spaceof singular anti-self-dual connections QM.S4; S2; P; '/ on the one hand, and onthe other, the set of isomorphism classes of collections .P ; ; �/ where

� P ! CP2 is a holomorphic principal Gc-bundle topologically isomorphicto ��.P /,

� W † ! OP is a holomorphic section of the associated bundle on † DCP1 with fiber O, homotopic to the section ��.'/.

� � is a holomorphic trivialization of the restriction of P to `1, satisfyingthe constraint that the induced trivialization of the adjoint bundle carries .s1/ to ˆ.

A special case of this theorem, which may help to understand the state-ment, is the case that k D 0 and the bundle P on S4 is trivial. In this case P

is topologically trivial; and the trivialization on the line at infinity forces P tobe analytically trivial also, so that � extends uniquely to a holomorphic triv-ialization of P ! CP2. The data then becomes a based rational map: aholomorphic map from † D CP1 to O sending s1 to ˆ.

Staying with this special case, the inequalities of Proposition 2.10 have astraightforward interpretation. For a holomorphic map from CP1 toO, thepairing of .CP1/ with any class in the closure of the Kahler cone of O mustbe non-negative. The inequalities of the proposition when k D 0 can be seen as

35

consequences of this statement. This is essentially the same argument that wasused by Murray [28] to constrain the possible charges of monopoles on R3.

For the general case, the strategy is similar, but we use the energy E of theanti-self-connection, rather than the energy of a holomorphic map. The essen-tial point, which the following immediate corollary of Theorem 2.13 above:

Corollary 2.14. Suppose ˆ and ˆ1 are two elements of the fundamental Weylchamber with the same stabilizer, so that z.Gˆ/ D z.Gˆ1/. Suppose both satisfythe constraint (9). Then M.S4; S2; P; '/ is homeomorphic to M.S4; S2; P; '1/when ' and '1 are sections of the bundles associated to the adjoint action ofG onthe orbits ofˆ1 andˆ2 respectively, with the same homotopy class. In particular,one of these moduli spaces is non-empty if and only if the other is.

To apply this corollary, suppose that a moduli space M.S4; S2; P; '/ isnon-empty. Let k 2 Z and l 2 L.Gˆ/ � z.Gˆ/ be the topological invariants ofP and '. Let A be the alcove in t defined as the intersection of the fundamentalWeyl chamber with the half-space � � 1, where � is the highest root. Thus A

is a closed simplex. The intersection A \ z.Gˆ/ is a simplex with possiblysmaller dimension. In applying the corollary, the admissible values for ˆ1 areprecisely the interior points of the simplex A \ z.Gˆ/. A necessary conditionfor a moduli space to be non-empty is that the associated topological energyE.S4; S2; P; '1/ is non-negative, so the corollary tells us that

2h_k C h ; li � 0 (44)

for all interior points of A \ z.Gˆ/, and hence for all points in the closed sim-plex A\ z.Gˆ/, by continuity. If … W t! z.Gˆ/ again denotes the orthogonalprojection, then it is a fact about the geometry of A that

….A/ � A \ z.Gˆ/:

As l itself lies in z.Gˆ/, we deduce that the inequality (44) holds not just for in A \ z.Gˆ/, but for all in A.

The vertices of the simplex A are the point 0 and the points D w�˛=�.w�˛/,

as ˛ runs through the simple roots. Applying (44) with at these vertices, weobtain k � 0 and

2h_�.w�˛/k C w˛.l/ � 0

for all simple roots ˛. To complete the proof of the inequality (42), we calcu-

36

late, using (6) and the definition of the coroots,

2h_�.w�˛/ D 2h�; �i�1w˛.�

�/

D w˛.�_/

D n_˛:

This completes the proof of the proposition.

Example. We illustrate the SU.N / case. Arrange the eigenvalues ofˆ as usual,as i�1, . . . i�m with �1 > � � � > �m and �1��m < 1. LetNs be the multiplicityof the eigenspace for �s, so that ' defines a reduction of P jS2 to the subgroup

S.U.N1/ � � � � � U.Nm//:

Let k be c2.P /ŒS4�, and let l1; : : : ; lm be the first Chern numbers,

ls D �c1.Es/ŒS2�

whereEs is the associated U.Ns/ bundle. We havePls D 0. Then the inequal-

ities of Proposition 2.10, taken just for the extreme cases when ˛ is in SC.ˆ/,become

k � 0

k C l1 � 0

: : :

k C l1 C l2 C � � � C lm�1 � 0:

(45)

Note that the first inequality can also be written as the non-negativity of k Cl1 C � � � C lm, because the ls add up to zero. Let us write

Ks D k CXt<s

lt ;

so that the above inequalities assert Ks � 0. Then we observe that the for-mal dimension of the framed moduli space, given by the formula (33), can bewritten as

2.Nm CN1/K1 C 2.N1 CN2/K2 C � � � C 2.Nm�1 CNm/Km:

This is bounded below by

4.K1 C � � � CKm/:

In particular the dimension of the moduli space is at least 4, unless k and thels are all zero. Slightly more precisely, we can state:

37

Corollary 2.15. For ˆ as above, and G D SU.N /, the minimum possible formaldimension of any non-empty framed moduli space of positive formal dimensionon .S4; S2/ is

minf 2.Ns�1 CNs/ j s D 1; : : : ; m g

where we interpret N0 as a synonym for Nm. In particular, no moduli spacehas dimension less than 4, except for the trivial zero-dimensional moduli space;and in the special case that there are only two distinct eigenvalues, the smallestpositive-dimensional moduli space has dimension 2N .

2.8 Orbifold metrics and connections

Up until this point, we have considered a moduli space M.X;†;P; '/ of sin-gular instantons defined using a space of connections Cp.X;†;P; '/modeledon an Lp1 Sobolev space, with p a little bigger than 2. There are disadvantagesassociated with having to use such a weak Sobolev norm: for example, theseconnections A are not continuous, which creates difficulties if we want to useholonomy perturbations later. There is also a difficulty with proving the sortof vanishing theorems that are usually used to show that the moduli spaces ofsolutions on S4, for example, are smooth.

Something that was exploited in [20] is that we can use stronger Sobolevnorms if we first make a slight change to the geometry of our picture. We willexplain this here.

We shall equip X with a singular metric g� which has an orbifold-typesingularity along the surface †, with cone-angle 2�=� for some integer � >0. This means that at each point of † there is a neighborhood U such that.U n†; g�/ is isometric to the quotient of a smooth Riemannian manifold by acyclic group of order �: the model for such a metric in the flat case is the metric

du2 C dv2 C dr2 C

�r2

�2

�d�2:

As motivation, ifˆ is an element of t � g with the property that �ˆ is integral,then our model singular connection A' from (13) can be constructed so thatit becomes a smooth connection on passing to �-fold branched cover; so if weuse the metric g� , then we can regard A' as an orbifold connection, and wecan reinterpret it as being a smooth connection in the orbifold sense.

We shall not use the orbifold language here, except in referring to the met-ric g� as having an “orbifold singularity”. Also, when using the metric g� , weshall not require that �ˆ be integral. What we will exploit is that, by mak-ing � sufficiently large, the “Fredholm package” that is used in constructing

38

the moduli spaces can be made to work in Sobolev spaces with any desireddegree of regularity. More precisely, let A' be the model singular connectionon .X;†/ equipped with the metric g� , and let dCA' be the linearized anti-self-duality operator acting on gP -valued 1-forms, defined using the metric g� .On differential forms on Xn†, define the norms LLp

k;A'using the Levi-Civita

derivative of g� and the covariant derivative of A' on gP . Then let D' be theoperator

D' D �d�A' ˚ d

C

A' (46)

acting on the spaces

LLp

k;A'.Xn†;gP ˝ƒ

1/! LLp

k�1;A'.Xn†;gP ˝ .ƒ

0˚ƒC// (47)

Then D' is Fredholm, as shown in [20, Proposition 4.17]:

Proposition 2.16 ([20]). Given any compact subinterval I � .0; 1/ and any p andm, there exists a �0 D �0.I; p;m/ such that for all � � �0, all k � m and all ˆin the fundamental Weyl chamber satisfying

˛.ˆ/ 2 I;8˛ 2 RC.ˆ/;

the operator D' acting on the spaces (47) is Fredholm, as is its formal adjoint,and the Fredholm alternative holds.

This proposition gives us the linear part of the theory needed for the gaugetheory; the non-linear aspects are the multiplication theorems and the Rellichlemma, which also go through in this setting: see [20] for details. When usingthe orbifold-type metric, we will fix an integer m > 2 and define our space ofconnections as

C.X;†;P; '/ D fA j A � A' 2 LL2m;A' g:

We write G .X;†;P; '/ for the corresponding gauge group, whose Lie algebrais L2mC1;A' .Xn†;gP /, and we let

M.X;†;P; '/ � C.X;†;P; '/=G

be the moduli space of singular anti-self-dual connections for the metric g� .The formula for the dimension of the moduli space at an irreducible regularpoint (i.e. the index of the operator D' above) is given by the same formula(19) as before, as is the energy E of a solution.

39

2.9 Orienting moduli spaces

We next show that the moduli spaces of singular instantons are orientable, anddiscuss how to orient them. Again, for the case G D SU.2/, the necessarymaterial is in [20]. In the case that the K is absent, the orientability of themoduli spaces for a general simple Lie group G and simply-connected X isexplained in [9]. For the case of SU.N / and arbitrary X , a proof is given in[7]. In the following proposition, we treat a simple, simply-connected group G.Recall that † is an oriented surface.

Proposition 2.17. In the moduli space M.X;†;P; '/, the set of regular pointsM reg is an orientable manifold. If the dimension of G is even, then M reg has acanonical orientation; while if G is odd-dimensional, the manifold M reg can becanonically oriented once a homology orientation for X is given.

Proof. We first deal with the case that† is absent: we consider the orientabilityof the set of regular points in a moduli space M.X;P / of (non-singular) anti-self-dual connections. Following the usual argument, we consider the space ofall connections modulo gauge, B.X; P /, and also the space of framed connec-tions QB.X; P /: the quotient of the space of connections by the based gaugegroup. Over QB.X; P / one has a real determinant line bundle �.X;P /, thedeterminant of the family of operators obtained by coupling �d� ˚ dC tothe family of connections in the adjoint bundle. To show that M reg.X; P / isorientable, we will show that �.X;P / is trivial.

We will reduce the problem to the known case of an SU.2/ bundle by ap-plying (in the reverse direction) the same stabilization argument used in [7].Pick any long root for G, say the highest root � , and let j W SU.2/! G be thecorresponding copy of SU.2/. The structure group of P can be reduced to thesubgroup j.SU.2//, giving us an SU.2/ bundle Q � P , and we have a map

j� W QB.X;Q/! QB.X; P /

whose domain is the space of based SU.2/ connections. From [7], we knowthat the corresponding line bundle �.X;Q/ on QB.X;Q/ is trivial.

The pair .G; j.SU.2// is 4-connected, so the inclusion of based gaugegroups is surjective on �0; and the map j� above is therefore surjective on �1.To show that the determinant line �.X;P / on QB.X; P / is trivial, it is there-fore enough to show that its pull back by j� is trivial. As stated in section 2.1,the adjoint representation of G decomposes as a representation of j.SU.2//as one copy of the adjoint representation of SU.2/, a number of copies of the2-dimensional representation of SU.2/, and a number of copies of the trivial

40

representation. Accordingly, the pull-back of �.X;P / by j� is a tensor prod-uct of a number of real line bundles: the one corresponding to the adjoint rep-resentation is a copy of �.X;Q/; the determinant lines corresponding to the2-dimensional representations are orientable using the complex orientations;and the remaining factors are trivial. Thus the triviality of �.X;P / is reducedto the known case of �.X;Q/.

Once one knows that the moduli space is orientable, the next issue is tospecify a standard orientation. We stay with the case that † is absent. By“addition of instantons”, the matter is reduced to specifying a trivializationof �.X;P / in the case that P is trivial. In this case we can look at the fiberof �.X;P / at the trivial connection, where operator is the standard opera-tor �d� ˚ dC coupled to the trivial bundle g. In the case that g is even-dimensional, the determinant line can be canonically oriented; while in the casethat g is odd-dimensional, we need to specify and orientation the determinantof the operator �d� ˚ dC with real coefficients, i.e. a homology orientationfor X . Conventions for these choices can be set up so that the orientation ofthe moduli space agrees with its complex orientation when X is Kalher: thearguments from [7] are adapted to the case of SU.N / in [19], and the case of ageneral simple G is little different.

We now consider how the determinant line changes when we introducea codimension-2 singularity along † � X . There is again a determinant linebundle�.X;†;P; '/ over the space QB.X;†;P; '/ of singular connections. Wemust show this line bundle is trivial. If we consider the restriction of the linebundle to a compact family S � QB.X;†;P; '/ of singular connections, thenthe data is a family of G bundles Ps with a family of reductions of structuregroup, 's. Let As be a family of smooth connections in the bundles Ps. Wemay suppose that Asj† respects the reduction 's to the group Gˆ � G. Wemay also suppose that As can be identified with pull-back of Asj† in a tubularneighborhood of †. Let A's be constructed from As by adding the singularterm in the usual way, as at (13). Over S , we can consider two line bundles:first the line bundle �.X;P /, which we have already seen is trivial; and secondthe line bundle�.X;†;P; '/, the determinant line of the deformation complexfor the singular instantons. We must examine the ratio

�.X;†;P; '/˝�.X;P /�1 (48)

and show that it is trivial.By excision, we can replace X now by the sphere-bundle over † which

is obtained by doubling the tubular neighborhood, and we can replace the

41

family of connections As by the family of Gˆ connections obtained by pullingback Asj†. In this setting, the adjoint bundle gP over S � † decomposes asa sum of two sub-bundles, associated to the decomposition of g as gˆ ˚ o in(10). There is a corresponding tensor product decomposition of each of thedeterminant lines in (48) above. On the summand gˆ, the two operators agree;so the ratio (48) is isomorphic to the ratio of determinant lines for the sameoperators coupled only to the subbundle coming from o instead of to all of g.Since o is complex, the ratio of determinant lines can be given its complexorientation, which completes the proof that the moduli space is orientable.This argument also shows that a choice of orientation for �.X;P / gives riseto a preferred orientation for �.X;†;P; '/. So the data needed to orient themoduli space is the same in the singular and non-singular cases.

2.10 The unitary and other non-simple groups

Up until this point, G has always been a simple and simply-connected group.One very straightforward generalization is allow G to be semi-simple and stillsimply-connected. In this case G is a product of simple groups G1 � � � � �Gm, and our configuration spaces of connections are simply products. We candefine the lattice L.Gˆ/ and the the monopole charge l just as before, with theunderstanding that we are now dealing with a root system that is reducible.The only new feature here is that the instanton charge k is now an m-tuple,k D .k1; : : : ; km/, and in the dimension and action formulae we seeX

h_i ki

with h_i the dual Coxeter number ofGi , where previously we had just h_k. Themonotone condition and Proposition 2.5 are unchanged.

There is a more interesting variation to consider when G has center of pos-itive dimension, such as in the case of the unitary group. We will make theassumption that G is connected and that the commutator subgroup ŒG;G� issimply connected. We shall write Z.G/ for the center of G, and we set

NZ.G/ D G=ŒG;G�

D Z.G/=.Z.G/ \ ŒG;G�/:

The quotient map will be written

d W G ! NZ.G/; (49)

42

and we use the same notation also for the corresponding map on the Lie alge-bras. The abelian group Z.G/ may not be connected, but NZ.G/ is a torus. Wewill run through some of the points to show how the theory adapts to this case.

Instanton moduli spaces. Let us temporarily omit the codimension-2 singular-ity along † from our discussion. An appropriate setting for gauge theory ina G-bundle P ! X when Z.G/ has positive dimension is not to considerthe space of all G-connections in P , but instead to fix a connection ‚ in theassociated NZ.G/-bundle, d.P / ! X , and to consider the space C.X; P / ofconnections A in P which induce the given connection ‚ in d.P /:

C.X; P / D fA j d.A/ D ‚ g

(In the case of the unitary group U.N/, this means looking at the unitary con-nections in a rank-N vector bundleE inducing a given connection‚ inƒNE.)Such a G-connection A in P is entirely determined by the induced connectionNA in the associated .G=Z.G//-bundle NP . The appropriate gauge group in this

context is not the group of all automorphisms of P , but instead the groupG .X; P / consisting of automorphisms which take values in ŒG;G� everywhere.That is, an element of G .X; P / is a section of associated fiber bundle arisingfrom the adjoint action of G on the subgroup ŒG;G�. In the case of a unitaryvector bundle, this is the group of unitary automorphisms of a vector bundleE ! X having determinant 1 at every point. The moduli space M.X;P / isthe subspace of the quotient B.X; P / D C.X; P /=G .X; P / consisting of allŒA� such that the curvature of NA (not the curvature of A) is anti-self-dual:

M.X;P / D fA 2 C.X; P / j FCNAD 0 g=G .X; P /:

Note that the chosen connection ‚ really plays no role here, and we couldequally well regard C.X; P / as paramatrizing the connections NA in NP . (Later,however, when we introduce holonomy perturbations in section 3.2, a choiceof ‚ will be important.) Indeed, B.X; P / and M.X;P / really depend only onthe adjoint group NG and the bundle NP , because both the adjoint bundle withfiber Œg;g� and the bundle of groups with fiber ŒG;G� (whose sections are thegauge transformations) are bundles associated to NP . The choice ofG thereforeonly affects which bundles NP can arise.

Because of this last observation, we can if we wish start with the simply-connected semi-simple group G1 with adjoint group NG1 and construct a groupG with ŒG;G� Š G1 as an extension. Such a G can be described by taking a

43

subgroupH1 of the finite groupZ.G1/ together with a torus S and an injectivehomomorphism a W H1 ! S ; one then defines

G D .G1 � S/=H (50)

where H � H1 � S is the graph of a. A given bundle NP with structure groupNG D NG1 lifts to a G-bundle P if and only if its characteristic class

Nc. NP / 2 H 2.X I�1. NG1//

lifts to a class

c 2 H 2.X I�1.G//:

The image of the map �1.G/! �1. NG1/ Š Z.G1/ is our chosen subgroup H1and �1.G/ is torsion-free; so the bundle NP has a lift to a G-bundle if and onlyif Nc. NP / lies in the subgroup H 2.X IH1/ and admits an “integer lift” to Zk forone (and hence any) presentation of H1 as

0! Zk ! Zk ! H1 ! 0:

This discussion shows us that, in order to allow the largest possible collectionof NG1-bundles to lift, and to avoid redundancy, we may impose the followingconditions.

Condition 2.18. In the construction of the non-semi-simple group G from G1in (50), we may require

(i) the subgroup H1 � Z.G1/ is the whole of Z.G1/;

(ii) the rank k of the torus S is chosen to be equal to the number of gen-erators in a smallest possible generating set of H1; or equivalently, theimage of H1 in S is not contained in any proper sub-torus.

These conditions mean that ifG1 is simple, thenG D G1 in the case of typeE8, F4 or G2 (the simply connected cases); while S will be a circle group in allother cases except D2r , where S will be a 2-torus. These conditions do notdetermine G uniquely, in general. For example, in the case that G1 D SU.N /,the condition allows that G is U.N/ but also allows G to be U.N/=Cm, whereCm is a cyclic central subgroup of order prime to N .

44

Singular instantons. We fix a maximal torus and a set of positive roots for G.Because G is not semi-simple, the fundamental Weyl chamber in t D Lie.T /is the product of the fundamental Weyl chamber for Œg;g� with z.G/ D

Lie.Z.G//. The bundle P is no longer trivial on the 3-skeleton of X , because�1.G/ is non-trivial. We can identify �1.G/ with the lattice

L.G/ � z.G/

obtained as the projection of the integer lattice in t. The bundle P has a 2-dimensional characteristic class which we write as

c.P / 2 H 2.X IL.G//: (51)

Now we introduce the codimension-2 singularity along a surface † � X .Fix an element ˆ in the Lie algebra g belonging to the fundamental Weylchamber and satisfying �.ˆ/ < 1, where � is the highest root. Let O � g

be the orbit of ˆ under the adjoint action. This lies in a translate of Œg;g�inside g consisting of all elements with the same trace as ˆ. Choose a section' of the associated bundleOP j†, so defining a reduction of the structure groupof P j† to the subgroup Gˆ.

Let ‚ again be a fixed connection in the associated bundle d.P / on X ,and let A0 be a G-connection with d.A0/ D ‚. Choose an extension of thesection ' to the tubular neighborhood, and define a singular G connection onthe restriction of P to Xn† by the same formula as in the previous case:

A' D A0 C ˇ.r/' ˝ � (52)

The induced connection on d.P / is

‚' D ‚C ˇ.r/d.'/˝ �: (53)

We define a space of G-connections modeled on A' and having the same in-duced connection on d.P /:

Cp.X;†;P; '/ D fA' C a j a;rA'a 2 Lp.Xn†I Œg;g�˝ƒ1.X// g:

Our gauge group will consist of ŒG;G�-valued automorphisms of P jXn†:

Gp.X;†;P; '/ D fg j rA'g;r2A'g 2 L

p.Xn†I ŒG;G�P / g:

For p sufficiently close to 2, as in (17), the moduli space M.X;†;P; '/ is de-fined as the quotient by Gp of the set of solutions to FC

NAD 0 in Cp.X;†;P; '/.

45

Monopole charges. We continue to write Z.Gˆ/ for the center of the commu-tant Gˆ, and L.Gˆ/ � z.Gˆ/ for the image of the integer lattice in t under theprojection… W t! z.Gˆ/. The structure group of the bundle P' ! † can stillbe reduced to the subgroup T , and the resulting T -bundle is classified by anelement � in the integer lattice. The element l D ….�/ determines P' up to iso-morphism. The bundle P itself may be non-trivial on †, and l is constrainedby the requirement that

d.l/ D hc.P /; Œ†�i

in L.G/ Š �1.G/.

Dimension and energy. The formula for the formal dimension of the modulispace can be written in the same was as before (see (19)), except for the terminvolving the 4-dimensional characteristic class:

� 2p1.gP /ŒX�C 4�.l/C.dimO/

2�.†/ � .dimG/.bC � b1 C 1/ (54)

Note that the term �.l/ depends only on the projection of l into Œg;g�. Theterm involving p1.gP / satisfies a congruence depending on the associatedNZ.G/-bundle d.P /, via the 2-dimensional characteristic class c.P /. In the case

that the structure group of P reduces to the maximal torus, we obtain a lift ofc.P / to a class Oc 2 H 2.X IL.T //, and we then have

�2p1.gP / D �hOc; Oci

where the quadratic form on the right is defined using the semi-definite Killingform on the lattice L.T / and the cup-square on X . Modulo 4h_, the quan-tity on the right depends only on the image of c.P / in the finite groupH 2.X I�. NG//. Furthermore, in the case that ŒG;G� is simple, if P is alteredon a 4-cell in X , then p1.gP / changes by a multiple of 2h_ (as in the case ofthe 4-sphere); and since a general P can be reduced to the maximal torus onthe complement of a 4-cell, we have in general

�2p1.gP /ŒX� D �hOc; OciŒX� .mod 4h_/

where Oc is any lift of c.P / to H 2.X IL.T //. In the case of the unitary groupU.N/, we identify c.P / as the first Chern class, and the formula becomes

� 2p1.gP /ŒX� D �2.N � 1/c21.P /ŒX� .mod 4N/: (55)

46

The appropriate definition of the energy E is still the formula (20), with theunderstanding that the norm defined by the Killing form is now only a semi-norm, so that the formula for the energy actually involves only NA. In a similarmanner, the formula (22) now becomes

E D 8�2��2p1.gP /ŒX�C 2hˆ; li � hˆ;ˆi.† �†/

�; (56)

where the inner products are now only semi-definite.In these two formulae, as elsewhere, the component ofˆ in the center of g is

immaterial. The monotone condition (i.e. the condition that the terms in (54)and (56) which are linear in p1.g/ and l are proportional) is a constraint onlyon the component N which lies in Œg;g�. In formulae, the monotone conditionstill amounts to requiring that

hˆ; li D 2�.l/ (57)

for all l in z.Gˆ/. Given any ˆ0 in the fundamental Weyl chamber, there isa unique ˆ satisfying this condition with the additional constraints that (i)ˆ0 and ˆ have the same centralizer and (ii) ˆ0 and ˆ have the same centralcomponent in z.G/.

Isomorphic moduli spaces. For the following discussion, we return temporarilyto considering the moduli spaces M.X;P / of non-singular connections, in theabsence of the embedded surface †. If P and P 0 are isomorphic G-bundles,then the moduli spaces M.X;P / and M.X;P 0/ are certainly homeomorphicalso; but a particular identificationM.X;P /!M.X;P 0/ depends on a choiceof bundle isomorphism f W P ! P 0. Because we are dividing out by the actionof the gauge group G .X; P / consisting of all ŒG;G�-valued automorphisms, themap of moduli spaces M.X;P / ! M.X;P 0/ depends on f only through thecorresponding isomorphism of NZ.G/-bundles, d.f / W d.P /! d.P 0/.

A convenient viewpoint on this is to fix a principal NZ.G/-bundle ı on X ,together with a connection ‚ on ı, and then regard B as parametrizing iso-morphism classes of triples consisting of:

(i) a principal G-bundle P ! X with specified instanton charges k D.k1; : : : ; km/;

(ii) an isomorphism of G-bundles q W d.P /! ı;

(iii) a connection A in P with d.A/ D q�.‚/, or equivalently just a connec-tion NA in NP .

47

Two such triples .P; q; NA/ and .P 0; q0; NA0/ are isomorphic if there is an isomor-phism of G-bundles, f W P ! P 0, with f �. NA0/ D NA and q D q0 B d.f /. Fromthis point of view, it is natural to write the configuration space as Bk.X/ı , in-dicating its dependence on k and the NZ.G/-bundle. The corresponding modulispace can be written

Mk.X/ı � Bk.X/ı

It is clear that the automorphisms g W ı ! ı act on Bk.X/ı , preservingthe moduli space, by .P; q; NA/ 7! .P; g B q; NA/. Furthermore the action of gon Bk.X/ı is trivial if and only if g D d.f / for some bundle isomorphismf W P ! P with f �. NA/ D NA for all A. This last condition on f requires thatf take values in Z.G/. Thus the group which acts effectively is the quotientof Map.X; NZ.G// by the image of Map.X;Z.G// under the map d W Z.G/ !NZ.G/. This is a finite group. For example, in the case of U.N/, this is the

quotient of H 1.X IZ/ by the image of multiplication by N ; this is isomorphicto the subgroup of H 1.X IZ=N/ consisting of elements with an integer lift.

There is another way in which isomorphisms arise between moduli spacesof this sort. The group operation provides a homomorphism of groups,

G �Z.G/! G:

Given a G-bundle P and a Z.G/-bundle � on X , we can use this homomor-phism to obtain a “product” G-bundle, which we will denote by P ˝ �. If wefix a connection ! in �, then to each connection A in C.X; P / with d.A/ D ‚,we can associate a connection

A0 D AC !

in C.X; P˝�/with d.A0/ D ‚Cd.!/. This operation descends to the quotientspace B and preserves the locus of connections which satisfy the equationsFCNAD 0. It therefore gives an identification of moduli spaces

�� WM.X;P /!M.X;P ˝ �/:

In terms of the data k and ı which determine the moduli space up to isomor-phism, this is a map

�� WMk.X/ı !Mk.X/ı˝d.�/;

where we have extended the use of ˝ to denote also the product of two NZ.G/-bundles. (In the case of U.N/, the operation ˝ in both cases becomes thetensor product by a line bundle.)

48

All of this discussion can be carried over to the case of connections withsingularities along † � X . Once ˆ is given, the moduli space M.X;†;P; '/is determined up to isomorphism by ı D d.P / together with the instantoncharges k and monopole charges l . (In the case that† has more than one com-ponent, the monopole charges need to be specified for each component.) Wecan therefore write the moduli space as Mk;l.X;ˆ/ı . The automorphisms ofı then act on the moduli spaces, and it is again the quotient of Map.X; NZ.G//by the image of Map.X;Z.G// that acts effectively. If � is a Z.G/-bundle, wealso have a corresponding isomorphism

�� WMk;l.X;ˆ/ı !Mk;l.X;ˆ/ı˝d.�/:

Reducibles. The main point at which the the present discussion of non-semi-simple groups diverges from the previous case is in the discussion of reducibleconnections, because the characteristic class c.P / now plays a role and becausethe integer lattice is no longer generated by the coroots. Let us write L.T / forthe integer lattice of the maximal torus; the lattice of weights is the dual latticein t�. The simple coroots ˛_ are a basis for L.T / \ Œg;g�, because ŒG;G� issimply connected. Let Nw˛ denote the dual basis for .t \ Œg;g�/�. For eachsimple root ˛, we can choose a weight w˛ in t� such that the restriction of w˛to t \ Œg;g� is Nw˛. The choice of w˛ is uniquely determined by Nw˛ to withinthe addition of a weight that factors through d. Associated to ˛, as before, is asubgroup G.˛/ � G, the centralizer of w�˛ (or equivalently of Nw�˛). Note thatwe will not always be able to choose w˛ in such a way that its restriction toz.G/ is zero, and nor will its restriction to the lattice L.G/ � z.G/ be integral:it will define a map

w˛ W L.G/! Q:

In the case of U.2/ for example, w˛ will map the rank-1 lattice L.G/ onto 12Z.

To say that ŒA� is reducible still means that its stabilizer in the gauge groupGp.X;†;P; '/ has positive dimension, and this is equivalent to there being anon-zero covariant-constant section of the associated bundle Œg;g�P � gP .As in subsection 2.6, we obtain a reduction of the structure group of P to asubgroup G‰, and we write P � P for this G‰-bundle. For some element �in the Weyl group, G�.‰/ is contained in G.˛/ for some simple root ˛, and itfollows that w˛ B � defines a non-central character of G‰:

s W G‰ ! U.1/:

49

Applying s to the connection A gives a U.1/ connection s.A/ whose curvatureFs.A/ is an Lp form defines a de Rham class

i

2�ŒFs.A/� D c1.s.P // �

rXjD1

.w˛ B �j /.ˆ/P:D:Œ†j � (58)

where the †j are the components of †, just as at (36). Unlike the previouscase, the curvature form Fs.A/ is not anti-self-dual, because FCA has a non-zerocentral component. The Chern-Weil formula for the central component is

i

2�ŒFd.A/� D c.P / �

rXjD1

d.ˆ/P:D:Œ†j �

as an equality of z.G/-valued cohomology classes. The self-dual part of FAcoincides with the self-dual part of Fd.A/. So, applying w˛ B � to this lastformula and then subtracting it from (58), we learn that the class

c1.s.P // � w˛.c.P // �

rXjD1

.w˛ B �j /.ˆ � d.ˆ//P:D:Œ†j �

is represented by an anti-self-dual form onX . The first term in this last formulais an integral class. The second term may not be integral: the class c.P / takesvalues in L.G/, and w˛ need not take integer values on this lattice. The termsin the last sum depend only on Nw˛, not on w˛, because ˆ � d.ˆ/ lies in Œg;g�.This leads to the following variant of Proposition 2.6.

Proposition 2.19. Suppose that bC.X/ � 1, and let the components of † be†1; : : : ; †r . Write

N D ˆ � d.ˆ/

for the component of ˆ in Œg;g�. Suppose that for every fundamental weight w˛and every choice of elements �1; : : : ; �r in the Weyl group, the real cohomologyclass

w˛.c.P //C

rXjD1

. Nw˛ B �j /. N /P:D:Œ†j �

is not integral. Then for generic choice of Riemannian metric on X , there are noreducible solutions in the moduli space M.X;†;P; '/.

Note that, as a rational cohomology class, w˛.c.P // does depend on w˛,not just on Nw˛. But different choices of how to extend Nw˛ will be reflected in achange to w˛.c.P // by an integral class.

50

In the case that each component †j is null-homologous, the criterion inthe above proposition reduces to the requirement:

w˛.c.P // be non-integral for each simple root ˛: (59)

To illustrate this, consider the familiar case of the unitary group U.N/. Thereare .N � 1/ simple roots, ˛k , k D 1; : : : ; N � 1, and if we write a typical Liealgebra element in u.N / as

x D i diag.�1; : : : ; �N /;

then we can choose w˛k so that

w˛k .x/ D �1 C � � � C �k :

Then c.P / is related to the usual first Chern class c1.P / in such a way that

w˛k .c.P // D .k=N /c1.P /:

So the criterion in the proposition is that the rational class .k=N /c1.P / shouldnot be integral, for any k in the range 1 � k � N � 1. This is equivalent torequiring that the evaluation of c1.P / on some integral homology class shouldbe coprime to N . We record this corollary, which is familiar from the case ofnon-singular instantons:

Corollary 2.20. LetG be the unitary group U.N/. Suppose that bC.X/ � 1, andthat all components of † are null-homologous. If there is an integral homologyclass in X whose pairing with c1.P / is coprime to N , then for generic choice ofRiemannian metric on X , there are no reducible solutions in the moduli spaceM.X;†;P; '/.

Remark. One might hope that there would be something analogous to thiscorollary in the case of other simply-connected groups with non-trivial center,but unfortunately, the case of the unitary group is rather special. If the com-mutator subgroupG1 D ŒG;G� is a simply-connected simple group of any typeother than Ar , there will always be a fundamental weight w˛ which takes inte-ger values on L.G/; so the criterion (59) cannot be met. This phenomenon isnoted in [34, Proposition 7.8].

51

Orientations. The discussion of orientations from section 2.9 adapts readilyto the case of non-simple groups G with ŒG;G� simply-connected. Recallthat Proposition 2.17 has two parts: the first is the assertion that the modulispaces are orientable, and the second involves specifying a canonical orienta-tion. Adapting the first part is routine. For the second task, we need to observethat the restriction of the homomorphism (49) to the maximal torus T � G

admits a right-inverse,e W NZ.G/! T

d B e D 1:(60)

Fix such an e once and for all. From our chosen NZ.G/-connection ‚ ind.P / we now obtain a G-connection e.‚/ on a bundle isomorphic to P , withd.e.‚// D ‚. This comes with a reduction of structure group (on the wholeof X ) to the maximal torus, and a fortiori to Gˆ. Adding the singular termalong † in the usual way, we obtain a distinguished singular connection A' .Because A' respects a reduction to the maximal torus, the adjoint bundle withfiber Œg;g� decomposes as a direct sum of a bundle with fiber t \ Œg;g� and acomplex vector bundle, and there is a corresponding decomposition of the op-erator (46) whose determinant line we wish to orient. The induced connectionon the first summand is trivial. As in the previous argument, we can now pro-ceed by making use of the complex orientation on the second summand andthe homology orientation oW for the first summand. In this way, the modulispaces M.X;†;P; '/ become canonically oriented at all regular points.

Let us write ı again for the NZ.G/-bundle d.P /, and so denote the modulispace by Mk;l.X;ˆ/ı as above. Recall that in this setting, the automorphismsof ı act on the moduli space. The naturality of the construction of the ori-entation means that the automorphisms of this moduli space arising in thisway are orientation-preserving diffeomorphisms on the regular part. A moreinteresting question arises if we ask whether the map

�� WMk;l.X;ˆ/ı !Mk;l.X;ˆ/ı˝d.�/ (61)

preserves orientation. The singularity along† plays no role in the answer here,and we could equally well consider Mk.X/ı instead. This is a question whichwas treated for G D U.2/ in [7], and the argument was adapted for U.N/ in[19]. The case of a general non-semisimple group is little different, and we shallsummarize the results.

We shall write G1 D ŒG;G� again, and we shall suppose that Z.G1/ isnon-trivial (for otherwise G is simply a product). We will also require that G1

52

is simple, and that G is obtained from G1 by the construction (50) and thatCondition 2.18 holds. Then we have the following result:

Proposition 2.21. Under the above assumptions, the map �� in (61) is orientationpreserving if the simple group G1 is any group other than E6 or SU.N /. In thecase of E6, the result depends on the choice of e; but e W U.1/ ! E6 may bechosen so that �� is orientation-preserving for all � and ı. In the case of SU.N /,if G is chosen to the standard U.N/ and e W U.1/! U.N/ is the inclusion in thefirst factor of the torus U.1/N in U.N/, then �� is orientation-preserving if N is0, 1 or 3 mod 4. If N is 2 mod 4 then �� is orientation-preserving if and only ifc1.�/

2ŒX� is even.

Proof. Both e.ı/ and e.d.�// are T -bundles on X . Let a and b be their re-spective characteristic classes in H 2.X IL.T //. The calculation for U.N/ from[19] adapts with little change to show that �� preserves or reverses orientationaccording to the parity of the quantity,�

1

2ha Y bi C

1

4hb Y bi C �.b/Y �.b/

�ŒX� (62)

in which ha Y bi denotes the pairing in H 4.X IZ/ obtained from the semi-definite Killing form on L.T / and the cup product on X , and �.b/ is to beinterpreted as an element of H 2.X IZ/. The quantity above plainly dependsonly on the images of a and b under the projection to Œg;g�. Let us write Na, Nbfor these projections (with the torsion parts of the cohomology dropped). Wehave

Na 2 H 2.X IL. NT1//=torsion � H 2.X I t1/

Nb 2 H 2.X IL.T1//=torsion � H 2.X I t1/(63)

where L.T1/ is the integer lattice for the maximal torus T1 in the simply-connected groupG1 D ŒG;G� and andL. NT1/ is the integer lattice for T1=Z.G1/(the maximal torus of the adjoint form of G1). Let h�;�i2 denote the innerproduct on t1 D Lie.T1/ normalized so that the coroots ˛_ corresponding tothe long roots ˛ for G1 have length 2. We then have

hx; yi D 2h_hx; yi2

where h_ is the dual Coxeter number of G1, so the formula (62) can berephrased as �

h_h Na Y Nbi2 Ch_

2h Nb Y Nbi2 C �. Nb/Y �. Nb/

�ŒX� (64)

53

The pairing h�;�i2 gives a map

L. NT1/ � L.T1/! Z

and its restriction to the coarser lattice L.T1/ is an even form. The Weyl vector� takes integer values on L.T1/, so each of the three terms in (64) is an integer.

Let p be the least common multiple of the orders of the elements ofZ.G1/:so p is 2 in the case of Bn, Cn, D2n and E7, is 4 in the case of D2nC1, is 3 inthe case of E6 and is nC 1 in the case of An. Condition 2.18 ensures that themap

d W Z.G/! NZ.G/

has the propertyd�.�1.Z.G// D p � �1. NZ.G//:

This condition tells us that Nb lies in

p �H 2.X IL. NT1//=torsion � H 2.X IL.T1//=torsion:

We writeNb D p Ne

Ne 2 H 2.X IL. NT1//=torsion:

We also exclude the An case from our discussion, because the result of theProposition for SU.N / is contained in [19]. We examine the parity of the threeterms in (64), beginning with the first term, the quantity

h_h Na Y Nbi2ŒX�:

This is even if h_ is even, and the remaining cases to look at (with the aboveexclusions in mind) are Bn for any n and Cn for n even. In both these cases, thepairing h�;�i2 takes only even values on L. NT1/ � L.T1/, so in all these casesthis term is even. The second term in (64) is also even if h_ is even. If h_ is odd,then p is even and the term can be expressed as

.h_p=2/h Ne Y Nbi2ŒX�:

As with the first term, we are dealing with Bn or C2m, and the pairing is evenby the same mechanism. The third term can be written

p2�. Ne/Y �. Ne/ŒX�:

The case An has been excluded, and in all other cases � takes integer values onL. NT1/. So �. Ne/ is an integral class. If p is even, then this term is therefore even.

54

The only case where p is odd is the case of E6. In the E6 case, one can checkthat there exists a coset representative Ne1 for a generator of L. NT1/=L.T1/ ŠZ=3 such that �. Ne1/ is an even integer. We can choose e so that its image isspanned by this representative, and with such a choice this third term is againeven.

3 Instanton Floer homology for knots

3.1 Configuration spaces and flat connections

Let Y be a closed, connected, oriented 3-manifold, and let K � Y be an ori-ented knot or link. Take a simple, simply-connected Lie group G, and letP ! Y be a principal G-bundle (necessarily trivial). Fix a maximal torusand a set of positive roots as before, and choose ˆ in the fundamental Weylchamber, satisfying the constraint (9). Let O � g be its orbit, and let ' bea section of the associated bundle OP along K, defining a reduction of thestructure group of P jK to the subgroup Gˆ. Any two choices of section ' arehomotopic: the only topological data is in the pair .Y;K/ and the choice of Gand ˆ.

We will equip Y with a Riemannian metric g� that is singular along K, aswe did in dimension 4 in subsection 2.8 above: the cone-angle will be 2�=�.To ensure sufficient regularity, we let I denote a compact interval in .0; 1/ con-taining ˛.ˆ/ for all roots ˛ in RC.ˆ/, and take � to be at least as large as theinteger �0.I; 2;m/ supplied by Proposition 2.16, with m a chosen Sobolev ex-ponent not less than 3. (We will impose further restrictions on � shortly.) Weconstruct a model singular connection B' on the restriction of P to Y nK, justas in the 4-dimensional case (see (13)), and we introduce the space of connec-tions

C.Y;K;ˆ/ D fB j B � B' 2 LL2m;B' g:

Here LL2k;B'

denote the 3-dimensional Sobolev spaces defined just as in subsec-tion 2.8. Because they are trivial, we omit P and ' from our notation. Thereis a gauge group

G .Y;K;ˆ/ D fg j g 2 LL2mC1;B' g

and a quotient space

B.Y;K;ˆ/ D C.Y;K;ˆ/=G .Y;K;ˆ/:

Two connections B and B 0 belonging to C.Y;K;ˆ/ are gauge-equivalent asG-connections on Y n K if and only if they differ by the action of an element

55

of G .Y;K;ˆ/. As in the 4-dimensional case, we call a connection B reducibleif its stabilizer has positive dimension.

The space of connections C.Y;K;ˆ/ is an affine space, and on the tangentspace TBC we define an L2 inner product (independent of B) by

hb; b0iL2 D

ZY

�tr.ad.�b/ ^ ad.b0//; (65)

Thus we are using the Killing form to contract the Lie algebra indices, andthe Hodge star on Y and the wedge product to contract the form indices. TheHodge star is the one defined by the singular metric g� . We define the Chern-Simons functional on C.Y; P;ˆ/ to be the unique function

CS W C.Y;K;ˆ/! R

satisfying CS.B'/ D 0 and having formal gradient (with respect to the aboveinner product)

.grad CS/B D �FB :

From this characterization, one can derive as usual the formula

CS.B' C b/ D˝�FB' ; b

˛L2C1

2

˝�dB'b; b

˛L2C1

3

˝�Œb ^ b�; b

˛L2: (66)

The Chern-Simons functional is independent of the choice of Riemannian met-ric on Y , as can be seen by rewriting this formula using (65).

The homotopy type of G .Y;K;ˆ/ is that of the space of maps g W Y ! G

with g.K/ � Gˆ. It follows that the the space of components of the gaugegroup is

�3.G/ � ŒK;Gˆ�

which is isomorphic toZ˚ L.Gˆ/

r (67)

where r is the number of components of the link K and L.Gˆ/ � z.Gˆ/ is thelattice of Definition 2.2. In particular, there is a preferred homomorphism

d W G .Y;K;ˆ/! Z˚ L.Gˆ/; (68)

where the map to the second factor is obtained by taking the sum over all com-ponents ofK. An alternative way to think of d is to use a gauge transformationg in G .Y;K;ˆ/ to form the bundle S1 �g P over S1 � Y , together with its re-duction S1 �g ' over S1 � K, defined by '. This data over .S1 � Y; S1 � †/

56

has an instanton number k and monopole charge, as in the previous section(see Definition 2.2 in particular). Then d.g/ can be computed as .k; l/.

The Chern-Simons functional is invariant only under the identity compo-nent of the gauge group. To express this quantitatively, let B 2 C.Y;K;ˆ/

be a connection, let g be a gauge transformation, and write d.g/ D .k; l/ 2

Z � L.Gˆ/. Then we have

CS.B/ � CS.g.B// D 4�2.4h_k C 2hˆ; li/:

For a path W Œ0; 1� ! C.Y;K;ˆ/, we define the topological energy astwice the drop in the Chern-Simons functional; so we can reinterpret the lastequation as saying that a path from B to g.B/ has topological energy

E D 8�2.4h_k C 2hˆ; li/:

This is a formula which is familiar also for the energy of a solution on anyclosed-manifold pair .X;†/ with † � † D 0. For a path which formallysolves the downward gradient-flow equation for the Chern-Simons functionalon C.Y;K;ˆ/, the topological energy coincides with the modified path energy,Z 1

0

�k P .t/k2 C k grad CS. .t//k2

�dt2:

From the definition of the Chern-Simons functional, it is apparent thatcritical points of CS are the flat connections in C.Y;K;ˆ/. The image of thecritical points in the quotient space B.Y;K;ˆ/ can be identified with the quo-tient by the action of conjugation of the space of all homomorphisms

� W �1.Y nK/! G (69)

with the property that the holonomy around each positively-oriented meridianof K is conjugate to exp.�2�ˆ/. We shall write

C � B.Y;K;ˆ/

for this set of critical points.Reducible critical points of the Chern-Simons functional can be ruled out

on topological grounds, by the following criterion, whose proof follows thesame line as the 4-dimensional version, Proposition 2.6.

57

Proposition 3.1. Let the components of K be K1; : : : ; Kr . Suppose that for ev-ery fundamental weight w˛ and every choice of elements �1; : : : ; �r in the Weylgroup, the real cohomology class

rXjD1

.w˛ B �j /.ˆ/P:D:ŒKj � (70)

is not integral. Then there are no reducible connections in the set of critical pointsC � B.Y;K;ˆ/.

Because the criterion in this proposition is referred to a few times, we giveit a name:

Definition 3.2. We will say that .Y;K;ˆ/ satisfies the non-integral conditionif the expression (70) is a non-integral cohomology class for every choice offundamental weight w˛ and Weyl group elements �1; : : : ; �r .

3.2 Holonomy perturbations

We will perturb the Chern-Simons functional CS by adding a term f : areal-valued function on C.Y / invariant under the action of the gauge groupG .Y;K;ˆ/. The type of perturbation that we use is essentially the same as thatused in [13], though similar constructions appear in [40, 7] and elsewhere.

Let q W S1 � D2 ! Y nK be a smooth immersion of a closed solid torus.Regard the circle S1 as R=Z and let s be a corresponding periodic coordinate.Let GP ! Y be the bundle with fiber G over Y whose sections are the gaugetransformations of P , and for each z in D2 let

Holq.�;z/.B/ 2 .GP /q.0;z/

be the holonomy of the connection B around the corresponding loop basedat q.0; z/. As z varies, we obtain in this way a section Holq.B/ of the bundleq�.GP / on the disk D2.

Next suppose we have an r-tuple of maps,

q D .q1; : : : ; qr/;

with qj W S1 �D2 ! Y nK an immersion. Suppose further that there is someinterval Œ��; �� such that the restriction of qj to Œ��; ���D2 is independent ofj :

qj .s; z/ D qj 0.s; z/; for all s with jsj � �: (71)

58

The pull-back bundles q�j .GP / are all canonically identified with each other onthe subset Œ��; �� � D2, and we can regard the holonomy maps as defining asection

Holq.B/ W D2 ! q�1 .GrP /

of the r-fold fiber-product of the bundle GP pulled back to D2. Pick anysmooth function

h W Gr ! R

that is invariant under the diagonal action of G by the adjoint action on the rfactors. Such an h defines also a function on q�1 .G

rP /. Let � be a non-negative

2-form supported in the interior of D2 and having integral 1, and define

f .B/ D

ZD2h.Holq.B//�: (72)

A function f of this sort is invariant under the gauge group action.

Definition 3.3. A cylinder function on C.Y;K;ˆ/ is a function

f W C.Y;K;ˆ/! R

of the form (72), determined by an r-tuple of immersions as above and a G-invariant function h on Gr .

Remark. In the transversality arguments that arise later, the important featureof the class of functions f obtained in this way is that they separate pointsin the quotient space B.Y;K;ˆ/, and also that they separate tangent vectorsat points where the gauge action is free. There is a slightly different class ofperturbations that one can use and which serves just as well in the case thatG D SU.N /: one can drop the requirement that qj D qj 0 on Œ��; ���D2, butinstead put a more restrictive condition on h, namely that it be invariant underthe action of G on each of the r factors separately. This alternative approachis laid out in detail in [8], where it is explained that such functions do separatepoints of B when G D SU.N /: the key point is the following lemma.

Lemma 3.4 ([8]). Suppose h1; : : : ; hm and h01; : : : ; h0m are elements of SU.N /

and suppose that for all wordsW , the elementsW.h1; : : : ; hm/ andW.h01; : : : ; h0m/

are conjugate. Then there is a u 2 SU.N / such that h0i D uhiu�1 for all i .

This lemma fails for other groups: this is essentially the observation of Dynkin[11], that two homomorphisms f1; f2 W H ! G between compact Lie groupsmay be linearly equivalent without being equivalent, where linear equivalence

59

means that ! B f1 is equivalent to ! B f2 for all linear representations ! of G.The approach we have taken here is the one used in [13], and works for anysimple G.

We examine the formal gradient of such a cylinder function with respectto our L2 inner product on the tangent spaces of C.Y;K;ˆ/. Let @jh be thepartial derivative of h along the j ’th factor: after trivializing the cotangentbundle of G using left-translation, we may regard this as a map

@jh W Gr! g�:

Using the Killing form, we can also construct the g-valued function .@jh/�.The G-invariance means that this also defines a map

.@jh/� W GrP ! gP :

Let Hj be the section of q�j .gP / on D2 defined by

Hj D .@jh/�.Holq.B//:

We extend Hj to a section of q�j .gP / on all of S1 �D2 by using parallel trans-port along the curves s 7! q.s; z/: the resulting section Hj has a discontinu-ity at s D 0 because the parallel transport around the closed loops may benon-trivial. The formal gradient of the cylinder function f , interpreted as agP -valued 1-form on Y nK, is then given by

�

� rXjD1

.qj /�.Hj�/�: (73)

Note that on Œ��; �� � D2 we can regard each Hj as a section of the samebundle q�1 .gP /, and while each Hj has a singularity at s D 0, the sum of theHj ’s does not, because of the G-invariance of h. The above 1-form is thereforecontinuous at q1.f0g �D2/.

Our connections are of class L2m away from the link K, and as in [40, 19]a short calculation shows that the section defined by the holonomy is of thesame class. So the gP -valued 1-form (73) is indeed in L2m. It is also supportedin a compact subset of Y nK, so it defines a tangent vector to the space ofconnections C.Y;K;ˆ/. As an abbreviation, let us write Cm for our space ofconnections modelled on LL2m;B' , and Tm for its tangent bundle. For k � m wehave the bundle Tk ! Cm obtained by completing the tangent bundle in theLL2k

norm. We will write V for the formal gradient of the cylinder function f .The following proposition details some of its properties.

60

Proposition 3.5. Let f be a cylinder function and let V be its formal gradient(73), regarded as a section of Tm over C.Y;K;ˆ/ D Cm. Then V has the follow-ing properties:

(i) The formal gradient V defines a smooth section, V 2 C1.Cm; Tm/.

(ii) For any j � m, the first derivativeDV 2 C1.Cm;Hom.Tm; Tm// extendsto a smooth section

DV 2 C1.Cm;Hom.Tj ; Tj //:

(iii) There is a constant K such that kV.B/kL1 � K for all B.

(iv) For all j , there is a constant Kj such that

kV.B/k LL2j;B'� Kj

�1C kB � B'k LL2

j;B'

�j:

(v) There exists a constant C such that for all B and B 0, and all p with 1 �p � 1, we have

kV.B/ � V.B 0/kLp � CkB � B0kLp :

In particular, V is continuous in the Lp topologies.

In order to work with a Banach space of perturbations f , we will considerfunctions f W C.Y;K;ˆ/! R which are obtained as the sum of a series

f D

1XiD1

aifi

where the ai are real and ffigi2N is some fixed countable collection of cylin-der functions. We need to consider whether such a sum is convergent. Morespecifically, we would like the series of gradients Vi D gradfi to converge to asection

V D

1XiD1

aiVi (74)

of the tangent bundle to Tm, and we would like the limit V to share with Vi theproperties detailed in the above proposition. In the first part of this proposi-tion, when we say that a section V belongs to C1, we do not imply that thenorm of the n’th derivative DnV jB is uniformly bounded independent of B.

61

But the norm is bounded by a function of kB �B'k LL2m;B'

: we have for each n

a continuous function hn.�/ such that

kDnV jB.b1; : : : bn/kL2l;B'� hn.kB � B

'k LL2

m;B'/

nYiD1

kbik LL2m;B'

:

A similar remark applies to the derivatives of DV in the norms that appearin the second part of the proposition. Because of this, given any countablecollection of cylinder functions ffigi2N , we can find constants Ci such thatthe series (74) converges whenever the sumX

Ci jai j

converges, and such that the limit V of the series is smooth.Before proceeding further, we shall choose a suitable countable collec-

tion of cylinder functions fi , sufficiently large to ensure that we can achievetransversality. For each integer r > 0, we choose a countable set of r-tuples ofimmersions q W S1 �D2 ! Y ,

.qr;j1 ; : : : ; qr;jr /; j 2 N;

satisfying (71) which are dense in the C 1 topology on the space of such r-tuplesof immersions. For each r , we also choose a collection of smooth G-invariantfunctions fhr

kgk2N on Gr which are dense in the C1 topology. Finally we

combine these to form a countable collection of cylinder functions

fj;k;r D hrk.q

r;j1 ; : : : ; qr;jr /:

Definition 3.6. Fix a countable collection of cylinder functions fi and con-stants Ci > 0 as above. Let P denote the separable Banach space of all realsequences � D f�igi2N such that the series

k�kPdefD

Xi

Ci j�i j

converges. For each � 2 P , let f� DP�ifi be the corresponding function

on C.Y;K;ˆ/, and letV� D

Xi

�iVi

be the formal gradient of f� with respect to the L2 inner product.

62

What our discussion has shown is that, for suitable choice of constants Ci ,the series (74) will converge and the limit will also have the properties of cylin-der functions that are given in Proposition 3.5. The next proposition recordsthis, together with the fact that the dependence of the estimates on � 2 P is asexpected:

Proposition 3.7. If the constants Ci in the definition of the Banach space P growsufficiently fast, then the family of sections V� of Tm satisfies the following con-ditions.

(i) The map.�; B/ 7! V�.B/

defines a smooth map V� 2 C1.P � Cm; Tm/.

(ii) For any j � m, the first derivative in the B variable, DV� 2 C1.P �Cm;Hom.Tm; Tm// extends to a smooth section

DV� 2 C1.P � Cm;Hom.Tj ; Tj //:

(iii) There is a constant K such that kV�.B/kL1 � Kk�kP for all � and B.

(iv) For all j , there is a constant Kj such that

kV�.B/kL2j;B'� Kj k�kP

�1C kB � B'kL2

j;B'

�j:

(v) There exists a constant C such that for all B and B 0, and all p with 1 �p � 1, we have

kV�.B/ � V�.B0/kLp � Ck�kP kB � B

0kLp :

(vi) The function f� whose gradient is V� is bounded on Cm.Y;K;ˆ/.

We will refer to a perturbation f D f� of the Chern-Simons invariantwhich arises in this way as a holonomy perturbation.

3.3 Elliptic theory and transversality for critical points

We fix a Banach space P parametrizing holonomy perturbations as above, andwe consider now the critical points of the perturbed Chern-Simons functionalCSC f� , for � 2 P . These critical points are the connections B in C.Y;K;ˆ/

satisfying� FB C V�.B/ D 0: (75)

63

These equations are invariant under gauge transformation, and we denote by

C� � B.Y;K;ˆ/

the image of the set of critical points in the quotient space. The familiar com-pactness properties of the space of flat connections modulo gauge transforma-tions extend to show that C� is compact: we have, more generally, the followinglemma.

Lemma 3.8. Let C� � P �B.Y;K;ˆ/ be the parametrized union of the criticalsets C� as � runs through P . Then the projection C� ! P is proper.

Proof. Suppose that ŒBi � belongs to C�i and �i converges to � in P . Theterms V�i .Bi / are bounded in L1, so the curvatures of the connections Bi arebounded inL1 also. By Uhlenbeck’s theorem, we may assume (after replacingthe Bi by suitable gauge transforms) that the connection forms Bi � B' are abounded sequence in LLp1 , for all p. Uhlenbeck’s theorem also allows us tofind a finite covering of Y by balls U˛ (or orbifold balls centered at points ofK) together with gauge transformations gi;˛ such that the connections Bi;˛ Dgi;˛.Bi / are in g�-Coulomb gauge with respect to some trivialization of P jU˛ .The connection forms Bi;˛ on U˛ are also bounded in LLp1 for all p, and thegauge transformations gi;˛ are bounded in LLp2 .

The equations satisfied by Bi;˛ on U˛ are

�dBi;˛ D � � ŒBi;˛ ^ Bi;˛� � gi;˛.V�i .Bi //

d�Bi;˛ D 0:

The terms ŒBi;˛ ^ Bi;˛� are bounded in LL21, because of the continuity of themultiplication LLp1 � LL

p1 !

LL21 for p > 3. The term V�i .Bi / is bounded in LL21,as is gi;˛.V�i .Bi // therefore. On a smaller ball U 0˛ � U˛, these equations there-fore give us an LL22 bound on Bi;˛. The bootstrapping argument now followsstandard lines: on smaller balls U 00˛ , the connections Bi;˛ are bounded in allLL2j norms, and after passing to a subsequence, the gauge transformations gi;˛can be patched together to form gauge transformations gi such that gi .Bi /converges in the LL2m topology.

For fixed � , the left-hand side of (75) defines a smooth section,

B 7! �FB C V�.B/

of the bundle Tm�1 ! Cm. Because of the gauge invariance of the func-tional, this section is everywhere orthogonal to the orbits of the gauge group

64

on C.Y;K;ˆ/, with respect to the L2 inner product. To introduce some nota-tion to express this, let us first write C�m D C�.Y;K;ˆ/ as usual for the subsetof C.Y;K;ˆ/ consisting of irreducible connections, and let us decompose therestriction of Tj to C�m as

Tj D Jj ˚Kj ; (76)

where Jj;B is the LL2j completion of the tangent space to the gauge-group orbitthrough B (i.e. the image of the map u 7! dBu) and Kj;B/ its L2 orthogonalcomplement in LL2j .Y IgP /. Thus Kj;B is the space of b in LL2j .Y IgP / satisfyingthe Coulomb condition,

d�Bb D 0:

On C�m, the decomposition (76) is a smooth decomposition of a Banach vectorbundle. The gauge invariance of our perturbations means that V� is a sectionthe summand Km � Tm over C�m.

Definition 3.9. We say that a solution B1 2 C�.Y;K;ˆ/ to the equation (75) isnon-degenerate if the section B 7! �FB C V�.B/ of the bundle Km�1 ! C�m istransverse to the zero section at B D B1.

The space of holonomy perturbations is sufficiently large to ensure that allcritical points will be non-degenerate for a suitably chosen � :

Proposition 3.10. There is a residual subset of the Banach space P such that forall � in this subset, all the critical points of the perturbed functional CSC f� inC�.Y;K;ˆ/ are non-degenerate.

Proof. For critical points whose stabilizer coincides with the center Z.G /, thisproposition follows from the fact that, given any compact (finite-dimensional)submanifold S of B�.Y;K;ˆ/, the functions f� jS are dense in C1.S/. Thisargument is just as in [13], [8] or [24]. For groups G other than SU.N /, wemust deal with the fact that irreducible connections may have finite stabilizerlarger than Z.G /. Our holonomy perturbations are still dense in C1.S/ forS a compact sub-orbifold of B�.Y;K;ˆ/ however, so this extra complicationcan be dealt with as in [41].

This says nothing yet about the reducible critical points; but we will beworking eventually with configurations .Y;K;ˆ/ satisfying the non-integralcondition of Definition 3.2, and we have the following version of Lemma 3.1for small perturbations of the equations:

65

Lemma 3.11. Suppose .Y;K;ˆ/ satisfies the non-integral condition. Then thereexists � > 0 such that for all � with k�kP � �, the critical points of CSC f� inC.Y;K;ˆ/ are all irreducible.

Proof. This follows from Lemma 3.1 and the compactness result, Lemma 3.8.

When a critical point B is non-degenerate, its gauge orbit ŒB� in C� is anisolated point of C� . It therefore follows that if � has norm less than � andbelongs also to the residual set described in Proposition 3.10, then C� is a finitesubset of B.Y / and is contained in B�.Y;K;ˆ/ D C�.Y;K;ˆ/=G .Y;K;ˆ/.We record this as a proposition.

Proposition 3.12. If .Y;K;ˆ/ satisfies the non-integral condition of Defini-tion 3.2, then there exists � > 0 and a residual subset of the �-ball in P , suchthat for all � in this subset the set of critical points C� in the quotient spaceB.Y;K;ˆ/ is a finite set and consists only of non-degenerate, irreducible criticalpoints.

Another way to look at the condition of non-degeneracy is to look at theoperator defined by the derivative of grad.CS C f�/ on C.Y;K;ˆ/, formallythe Hessian of the functional. This Hessian is a section

Hess 2 C1.Cm;Hom.Tm; Tm�1//

and is given byHessB.b/ D �dBb CDV jB.b/:

At a critical point B, the Hessian annihilates JB and maps KB to itself; andas an operator Kj;B ! Kj�1;B it is a compact perturbation of the Fredholmoperator �dB , because DVB maps LL2j to LL2j . As an unbounded self-adjointoperator on Kj;B it has discrete spectrum: the spectrum consists of eigenval-ues, the eigenspaces are finite-dimensional and the sum of the eigenspaces isdense. The non-degeneracy condition is the condition that HessB is invertible,or equivalently the condition that 0 is not an eigenvalue.

At a connection B that is not a critical point of the perturbed functional,the operator HessB does not leave invariant summands JB and KB ; and asan operator Tm;B ! Tm�1;B , it is not Fredholm. To correct this, one canintroduce the extended Hessian, which is the operator

bHessB D�0 �d�B�dB HessB

�

66

acting on the spaces

bHessB W LL2j .Y IgP /˚ Tj ! LL2j�1.Y IgP /˚ Tj�1:

This is a self-adjoint Fredholm operator which varies smoothly with B 2

C.Y;K;ˆ/; it is a compact perturbation of the family of elliptic operators�0 �d�B�dB �dB

�:

The extended Hessian also has discrete spectrum consisting of (real) eigenval-ues of finite multiplicity; the sum of the eigenspaces is again dense. At a criticalpoint B, the extended Hessian can be decomposed into the direct sum of twooperators, one of which is the restriction of HessB as an operator Kj !Kj�1.The other summand is invertible at irreducible critical points. It follows that,for a critical point B, the inevitability of bHessB is equivalent to the two condi-tions that B be both irreducible and non-degenerate.

In the case that the perturbation is zero, the set of critical points C �

B.Y;K;ˆ/ can be identified with a space of representations � of the funda-mental group of Y nK, as explained at (69). In this situation, a representation� determines a local coefficient system g� on Y nK, with fiber g. This has co-homology groups H i .Y nKIg�/. The following lemma (which is standard inthe absence of the knot K) provides a criterion for the non-degeneracy of thecorresponding connection B as a critical point of CS.

Lemma 3.13. In the above situation, the kernel of HessB on Kj is isomorphic to

ker W H 1.Y nKIg�/! H 1.mIg�/

where m is any collection of loops representing the meridians of all the compo-nents of K. A critical point is therefore non-degenerate if and only if the abovekernel is zero.

Proof. The kernel of the Hessian on Kj is isomorphic to ker.dB/=im.dB/ onour function spaces on Y with the orbifold metric. We can decompose Y asa union of two pieces, one of which is a tubular neighborhood of K and theother of which is the complement of a smaller neighborhood. The isomor-phism of the lemma then follows from a Mayer-Vietoris sequence, using thisdecomposition.

Suppose now that B0 and B1 are two irreducible, non-degenerate criticalpoints. Let B.t/ be a path in C.Y;K;ˆ/ from B0 to B1. We define

gr.B0; B1/ 2 Z

67

to be the spectral flow of the one-parameter family of operators bHessB.t/. Be-cause C.Y;K;ˆ/ is contractible, this number does not depend on the path, butonly on the endpoints. Now let

ˇ0 D ŒB0�; ˇ1 D ŒB1�

be the corresponding critical points in the quotient space B D B.Y;K;ˆ/.The path B.t/ determines a path � from ˇ0 to ˇ1. Let z 2 �1.B; ˇ0; ˇ1/ be therelative homotopy class of �. The homotopy class of z again depends only onB0 and B1. To turn this around, if ˇ0 and ˇ1 belong to C� � B.Y;K;ˆ/ andare both irreducible and non-degenerate, and if z is a relative homotopy classof paths from ˇ0 to ˇ1, we define

grz.ˇ0; ˇ1/ 2 Z

to be equal to gr.B0; B1/, where B0 and B1 are the endpoints of any pathwhose image in B.Y;K;ˆ/ belongs to the homotopy class z.

The fundamental group of B.Y;K;ˆ/ is equal to the group of componentsof G .Y;K;ˆ/, which is described as (67) earlier. If B is a non-degenerate,irreducible critical point, and if B 0 is obtained from B by applying a gaugetransformation g which is not in the identity component of G .Y;K;ˆ/, thena path from B 0 to B gives rise to a homotopy class of closed loops z based atthe corresponding point ˇ in B.Y;K;ˆ/. We can compute the spectral flowaround this loop in terms of the data .k; l/ D d.g/:

Lemma 3.14. Let B 0 D g.B/ in C.Y;K;ˆ/, and write

d.g/ D .k; l/ 2 Z � L.Gˆ/:

Then for the corresponding element z of �1.B.Y;K;ˆ/; ˇ/ obtained from a pathof connections from B to B 0, we have

grz.ˇ; ˇ/ D 4h_k C 4�.l/

where as usual h_ is the dual Coxeter number of G and � is the Weyl vector.

The proof of this lemma follows the expected line, reinterpreting the spec-tral flow of the family of operators as the index of an operator associatedto S1 � Y . The corresponding operator in this context is (a perturbationof) the linearized anti-self-duality equation with gauge fixing, so the index isthe formal dimension of a moduli space of singular instantons on the pair.S1 � Y; S1 �K/. This relationship between the Chern-Simons functional onC.Y;K;ˆ/ and singular instantons in dimension 4 is the subject of the nextsubsection.

68

3.4 The 4-dimensional equations and transversality for trajectories

Fix now some � 2 P , and write V for V� and f for f� . Let B.t/ be a path inC.Y;K;ˆ/, defined say on a bounded interval I � R. The path is a trajectoryfor the downward gradient flow of CSC f if it satisfies the equation

dB

dtD � � FB � V.B/:

The path B.t/ defines a connection A on the pull-back bundle over I � Y .This connection A is in temporal gauge (that is, it has no dt component whenexpressed in local trivializations obtained by pull-back), and it has a singularityalong I �K modelled on the singular connection A' obtained by pulling backB' . In terms of A, the above equation can be written

FCA C .dt ^ V.A//CD 0; (77)

where V.A/ denotes the 1-form in the Y directions obtained by applying V toeach B.t/, regarded as giving a 1-form on I �Y . In the form (77), the equationis fully gauge-invariant under the 4-dimensional gauge group.

As an abbreviation, let us write OV for the perturbing term here,

OV .A/ D .dt ^ V.A//C;

so that the equations areFCA C

OV .A/ D 0: (78)

This perturbing term for the 4-dimensional equations shares the same basicproperties as the perturbation V.B/ for the 3-dimensional equations. To statethese, we suppose the interval I is compact and write

Z D I � Y

L D I �K

so that L is an embedded 2-manifold with boundary in Z. We will continueto write P ! Z for what is strictly the pull-back of P from Y , and ' for thetranslation-invariant section of OP along L obtained by pulling back the sec-tion ' from K. We write Cm.Z;L;P; '/ for the space of connections singularconnections A D A'Ca with a of class LL2m. As an abbreviation, and to distin-guish it from the similar space of 3-dimensional connections Cm D C.Y;K;ˆ/,we write

C .4/m D C.Z;L;P; '/:

69

In a similar way, we write T.4/j for the LL2j completion of the tangent bun-

dle of C.4/m , and we write Sj ! C

.4/m for the (trivial) vector bundle with fiber

LL2j .ZIgP ˝ƒC.Z//. We assume from now on that m is at least 3, so that our

connections are again continuous on Z n L. Then we have the following factsabout OV , mirroring Proposition 3.7.

Proposition 3.15. Let A 7! OV .A/ be the perturbing term for the 4-dimensionalequations, regarded as a section of the bundle Sm over C

.4/m . Then

(i) The section OV is smooth:

OV 2 C1.C .4/m ;Sm/:

(ii) For any j � m, the first derivative

D OV 2 C1.C .4/m ;Hom.T .4/m ;Sm//

extends to a smooth section

D OV 2 C1.C .4/m ;Hom.T .4/j ;Sj //:

(iii) There is a constant K such that k OV .A/kL1 � K for all A.

(iv) For all j � m, there is a constant Kj such that

k OV .A/k LL2j;A'� Kj

�1C kA � A'k LL2

j;A'

�j:

(v) There exists a constant C such that for all A and A0, and all p with 1 �p � 1, we have

k OV .A/ � OV .A0/kLp � CkA � A0kLp :

In particular, OV is continuous in the Lp topologies.

In each of these cases, the dependence on � 2 P can also be included, as in thestatement of Proposition 3.7.

For a solution A in C.4/m on a compact cylinder Z D Œt0; t1� � Y , we define

the (perturbed) topological energy as twice the change in the functional CSCf� : that is,

E�.A/ D 2�.CSC f�/.B.t0// � .CSC f�/.B.t1//

�; (79)

70

where B.t/ is the 3-dimensional connection obtained by restricting A to ftg �Y . Because of the last condition in Proposition 3.7, the perturbing term hereonly affects the energy by a bounded amount. The Chern-Simons functionalis invariant only under the identity-component of the gauge group, so E�.A/

is not determined by knowing only the gauge-equivalences classes of the twoendpoints, ˇ0 D ŒB.t0/� and ˇ1 D ŒB.t1/�. The energy is determined by theendpoints ˇ0, ˇ1 in B.Y;K;ˆ/ and the homotopy class of the path z betweenthem given by ŒB.t/�. Accordingly, we may write the energy as

Ez.ˇ0; ˇ1/:

We turn next to the Fredholm theory for solutions to the perturbed equa-tions on the infinite cylinder. We write

Z D R � Y

L D R �K:

Let us suppose that the holonomy perturbation is chosen as in Proposi-tion 3.12, so that the critical points are irreducible and non-degenerate. Let˛ and ˇ be two elements of C� and z a homotopy class of paths betweenthem. Let B˛ and Bˇ be corresponding elements of C.Y;K;ˆ/, chosen so thata path from B˛ to Bˇ projects to B.Y;K;ˆ/ to give a path belonging to theclass z. Let A0 be a singular connection on the pull-back of P to the infinitecylinder Z, such that the restrictions of A0 to .�1;�T � and ŒT;1/ are equalto the pull-back of B˛ and Bˇ respectively, for some T . Define

Cz.˛; ˇ/ D fA j A � A0 2 LL2m;A0

.ZIgP ˝ƒ1.Z// g: (80)

This space depends on the choice of A0, not just on ˛, ˇ and z. But any twochoices are related by a gauge transformation. We define G .Z/ to be the groupof gauge transformations g of P on Z satisfying

g � 1 2 LL2mC1;A0.ZIGP /;

and we have the quotient space

Bz.˛; ˇ/ D Cz.˛; ˇ/=G .Z/:

It is an important consequence of the non-degeneracy of the critical points,that every solution A to the perturbed equations on R � Y which has finitetotal energy is gauge-equivalent to a connection in Cz.˛; ˇ/, for some ˛, ˇ andz.

71

Definition 3.16. The moduli space Mz.˛; ˇ/ � Bz.˛; ˇ/ is the space of gauge-equivalence classes of solutions to the perturbed equations, FCA C OV .A/ D 0.

Because we have assumed that all critical points are irreducible, the con-figuration space Cz.˛; ˇ/ consists also of irreducible connections. The actionof the gauge group therefore has only finite stabilizers, and Bz.˛; ˇ/ is a Ba-nach orbifold (or a Banach manifold in the case thatG D SU.N /): coordinatecharts can be obtained in the usual way using the Coulomb condition. Thelocal structure of Mz.˛; ˇ/ is therefore governed by the linearization of theperturbed equations together with the Coulomb condition: at a solution A inCz.˛; ˇ/, this is the operator

QA D �d�A ˚

�dCA CD

OV jA�

(81)

from LL2m;A0.ZIgP ˝ƒ1.Z// to LL2m�1;A0.ZIgP ˝ .ƒ

0 ˚ƒC/.Z//.This operator is Fredholm, and its index is equal to the spectral flow of the

extended Hessian:indexQA D grz.˛; ˇ/:

Definition 3.17. A solution A to the perturbed equations in Cz.˛; ˇ/ is regularif QA is surjective. We say that the moduli space Mz.˛; ˇ/ is regular if A isregular for all ŒA� in the moduli space.

If the moduli space is regular, then it is a (possibly empty) smooth orbifoldof dimension grz.˛; ˇ/. At this point, one would like to argue that for a genericchoice of perturbation � , all the moduli spacesMz.˛; ˇ/ are regular. However,although such a result is true for the case of SU.2/ (and is proved in [13] and[8]), the presence of non-trivial finite stabilizers is an obstruction to extend-ing the transversality arguments to general simply-connected simple groupsG.When the stabilizers are all equal to the center Z.G /, then the arguments fromthe SU.2/ case carry over without change. We therefore have:

Proposition 3.18. Suppose that �0 is a perturbation such that all the criticalpoints in C�0 are non-degenerate and have stabilizer Z.G /. Then there exists� 2 P such that:

(i) f� D f�0 in a neighborhood of all the critical points of CSC f�0;

(ii) the set of critical points for these two perturbations are the same, so C� D

C�0;

(iii) for all critical points ˛ and ˇ in C� and all paths z, the moduli spacesMz.˛; ˇ/ for the perturbation � are regular.

72

In order to proceed, we will require non-degeneracy for all critical pointsand regularity for all moduli spaces. We therefore impose the following condi-tions:

Hypothesis 3.19. We will assume henceforth that the triple .Y;K;ˆ/ satisfiesthe non-integral condition, and that a small perturbation � is chosen as inProposition 3.12 so that the critical points are irreducible and non-degenerate.We assume furthermore that the stabilizer of each critical point is just Z.G /,and that the moduli spaces Mz.˛; ˇ/ are all regular, as in the previous propo-sition.

In practice, we do not know how to ensure the condition in Hypothesis 3.19that the stabilizers be Z.G / except by taking G D SU.N /, in which case it isautomatic, given the other conditions. Of course, for any given .Y;K;ˆ/, it isalways possible that this condition is satisfied, as it were, “by accident”; butfrom this point on we really have SU.N / in mind. The notation we have set upfor a general simply-connected simple group G is still appropriate, and we willcontinue to use it.

3.5 Compactness and bubbles

The basic compactness results for singular instantons on a compact pair.X;†/, which we summarized in Proposition 2.9, can be adapted to the case ofsolutions on a compact cylindrical pair

Z D I � Y

L D I �K:

The main differences from the closed case are the following. First, in the caseof a closed manifold, the energy E is entirely determined by the topology of Pand '. In the case of a finite cylinder, the energy depends on the (perturbed)Chern-Simons invariants of the restriction of the connection to the two bound-ary components, and is therefore not constrained by the topology: in order toobtain a compactness results we need to impose a bound on the energy as partof the hypotheses. Second, since the proofs ultimately depend on interior esti-mates, the hypothesis of bounded energy for a sequence of solutions on Z willonly ensure that we have a subsequence converging on some interior domain.Third, when bubbles occur, their effect is no longer local, because of our non-local holonomy perturbations. None of these issues are special to the case of

73

instantons with singularities: they all occur in the standard construction of in-stanton Floer homology, and the issues surrounding the non-local holonomyperturbations are treated in [8] and [19].

In the statement of the following proposition (which corresponds to thefirst parts of Proposition 2.9), the LLp

ktopology refers to the topology on sec-

tions of gP ˝ƒi defined by using the covariant derivative of A' and the Levi-

Civita derivative of the orbifold metric g� , just as LL2k

was defined earlier.

Proposition 3.20. Let .Z;L/ be the compact cylindrical pair defined above, andlet I 00 � I be a compact sub-interval contained in the interior of I . Let An bea sequence of solutions to the perturbed equations in C.Z;L;P; '/, and supposethere is a uniform bound on the energy:

E�.An/ � C; for all i :

Then after passing to a subsequence, we have the following situation. There is aninterval I 0 with I 00 � I 0 � I , a finite set of points x contained in the interior ofthe sub-cylinderZ0 D I 0�Y , and a solutionA to the equations in C.Z0; L0; P; '/,with the following properties.

(i) There is a sequence of isomorphisms of bundles gn W P jZ0nx ! P of classLL2mC1 such that

g�n.An/! AjZ0nx

in the LLp1 topology on compact subsets of Z0nx for all p > 1.

(ii) In the sense of measures on Z0, the energy densities 2jFAn j2 converge to

2jFAj2C

Xx2x

�xıx

where ıx is the delta-mass at x and �x are positive real numbers.

The reason for passing from a subinterval I 00 to a larger one I 0 in the state-ment above is to ensure that the set of bubble-points x is contained entirely inthe interior of Z0. This means in particular that the gauge-transformations gnin the statement of the proposition are defined on the two boundary compo-nents of Z0. Let us write I 0 as Œt 00; t

01�, so that the boundary components are

ft 00g � Y and ft 01g � Y . Using again the map d defined at (68), we can considerthe elements

d.gnjt 0i/ 2 Z˚ L.Gˆ/

74

for i D 0; 1. If x were empty, then these two would be equal, but in generalthe difference is a topological quantity accounted for by the failure of gn toextend over the punctures. This is the same phenomenon that accounts for thedifference between .k; l/ and .k0; l 0/ in item (iii) of Proposition 2.9. CombiningProposition 2.9 with Proposition 2.10, we therefore obtain:

Proposition 3.21. In the situation of Proposition 3.20, we can choose the subse-quence so that the elements d.gnjt 0

i/ are independent of n for i D 0; 1; and for

each x 2 x, we can find .kx; lx/ 2 Z˚ L.Gˆ/ such that

d.gnjt 00/ � d.gnjt 01

/ D�Xx2x

kx;Xx2x

lx

�:

(If x does not lie on the surface L0 D I 0 � K, then lx is zero.) The energy �xthat is lost at x is then given by

�x D 8�2�4h_kx C 2hˆ; lxi

�:

Furthermore, the pairs .kx; lx/ are subject to the constraints of Proposition 2.10,namely

kx � 0; and

n_˛kx C w˛.lx/ � 0

for all simple roots ˛.

As in the case of a closed manifold, the energy lost at the bubbles is ac-counted for by solutions on the pair .S4; S2/ (equipped now with a conformally-flat orbifold metric as in [20]).

The compactness results above, for solutions on a compact cylinder, leadin a standard way to compactness results for solutions on the infinite cylinderR � Y when transversality hypotheses are assumed, as in Hypothesis 3.19. Tointroduce notation for this, if z is not the class of a constant path at ˛ D ˇ,we let MMz.˛; ˇ/ denote the quotientMz.˛; ˇ/=R, where R acts by translations.(For ˛ D ˇ and z the constant path, we regard MMz.˛; ˇ/ as the empty set.)We call the elements of MMz.˛; ˇ/ the unparametrized trajectories. By a broken(unparametrized) trajectory from ˛ to ˇ, we mean a collection

ŒAi � 2 MMzi .ˇi�1; ˇi /

for i D 1; : : : ; l , with ˇ0 D ˛ and ˇl D ˇ. The case l D 0 is allowed. Wewrite MMCz .˛; ˇ/ for the space of all unparametrized broken trajectories from ˛

75

to ˇ with the additional property that the composite of the paths zi is in thehomotopy class z.

In finite-dimensional Morse theory, the spaces of broken trajectories ofthis sort are compact. For the instanton theory, compactness holds only insituations where we can rule out the possibility that bubbles may occur. Givenour transversality hypotheses, we can rule out bubbles on the grounds of thedimension of the moduli spaces involved. In particular, from Corollary 2.12,we deduce:

Proposition 3.22. If the dimension of Mz.˛; ˇ/ is less than 4, then the spaceof unparametrized broken trajectories MMCz .˛; ˇ/ is compact. In particular, ifgrz.˛; ˇ/ D 1, then MMz.˛; ˇ/ is a compact zero-dimensional manifold.

The bound of 4 in this proposition can be improved in particular cases,depending on the group G and the choice of ˆ. The correct condition in gen-eral is that grz.˛; ˇ/ is smaller than the smallest dimension of any positive-dimensional framed moduli space on .S4; S2/. See Corollary 2.15 for example.

There is a significant additional question that does not arise in the case thatK is absent. The compactness result that we have just stated concerns a singlemoduli space. There are only finitely many critical points, but for each pair.˛; ˇ/ there are infinitely many possibilities for z. WhenK is empty, �1.B.Y //is Z and grz.˛; ˇ/ is a non-constant linear function of z: the moduli space willbe empty when grz.˛; ˇ/ is negative, and one should expect the moduli spaceto be non-empty (and of large dimension) once grz.˛; ˇ/ becomes large. WhenK is present, �1.B.Y;K;ˆ// is larger, and knowledge of grz.˛; ˇ/ no longerdetermines z. There may be infinitely many non-empty moduli spaces, all ofthe same dimension. What we do have is a finiteness result when a bound onthe energy is known. Let us again write

Ez.˛; ˇ/ D 2�.CSC f�/.B˛/ � .CSC f�/.Bˇ /

�I

for the (perturbed) topological energy along a homotopy class of paths z. Thisis the energy for any solution in the moduli space Mz.˛; ˇ/. For a proof of thefollowing finiteness result, see [24, Proposition-something].

Proposition 3.23. Given any C > 0, there are only finitely many ˛, ˇ and zfor which the moduli space Mz.˛; ˇ/ is non-empty and has topological energy atmost C .

For the construction of the Floer homology, the important comparisonis between the topological energy Ez.˛; ˇ/ and the spectral flow, or relative

76

grading, grz.˛; ˇ/. We can look at the special case where ˛ D ˇ so that a liftof a path in the class z gives a path of connections on Y nK from B to B 0,where B 0 differs from B by a gauge transformation g 2 G .Y;K;ˆ/. We writeB 0 D g.B/. Let us again set

d.g/ D .k; l/ 2 Z � L.Gˆ/

as in (68). Then for the corresponding homotopy class z of closed paths basedat ˇ D ŒB�, we have

grz.ˇ; ˇ/ D 4h_k C 4�.l/

andEz.ˇ; ˇ/ D 8�

2�4h_k C 2hˆ; li

�:

These formulae can be computed, for example, by applying the dimension andenergy formulae for the closed manifold S1 � Y containing the embedded sur-face S1 � K. In the case that ˆ satisfies the monotone condition (Defini-tion 2.4), these two linear forms in k and l are proportional. Since there areonly finitely many critical points in all, we see:

Lemma 3.24. If ˆ satisfies the monotone condition, then there is a constant C0such that for all ˛, ˇ and z, we haveˇ

Ez.˛; ˇ/ � 8�2 grz.˛; ˇ/

ˇ� C0:

From Proposition 3.23 we now deduce:

Corollary 3.25. If ˆ satisfies the monotone condition, then given any D > 0,there are only finitely many ˛, ˇ and z for which the moduli space Mz.˛; ˇ/ isnon-empty and has formal dimension at most D.

3.6 Orientations

If ˛ and ˇ are not necessarily critical points, we can still construct the operatorQA from an arbitraryA corresponding to a path � joining ˛ to ˇ. The operatoris Fredholm if the extended Hessian is invertible at both ˛ and ˇ. Under thesecircumstances, let us define

ƒ� .˛; ˇ/

to be the (two-element) set of orientations for the determinant line of the Fred-holm operator QA. As � varies in the paths belonging to a particular homo-topy class of paths z, the family of determinant lines of the corresponding

77

operators QA forms an orientable real line bundle over the space of paths: thisorientability can be deduced from the corresponding statement in the case ofa closed manifold, Proposition 2.17. An orientation for any one determinantline in this connected family therefore determines an orientation for any other.Thus it makes sense to write

ƒz.˛; ˇ/

in place of ƒ� .˛; ˇ/, for z a homotopy class of paths from ˛ to ˇ. If z0 is ahomotopy class of paths from ˇ to ˇ0, then there is a natural composition law,

ƒz.˛; ˇ/ �ƒz0.ˇ; ˇ0/! ƒz0Bz.˛; ˇ

0/:

(Note that our notation for a composite path puts the first path on the right.)Because of the requirement that the Hessian is invertible at the two end-points,the two-element set ƒz.˛; ˇ/ cannot be thought of as depending continuouslyon ˛ and ˇ in B.Y;K;ˆ/.

A priori, ƒz.˛; ˇ/ depends on z, not just on ˛ and ˇ; but we can spec-ify a rule, compatible with the composition law, that identifies ƒz.˛; ˇ/ andƒz0.˛; ˇ/ for different homotopy classes z and z0. This can be done, for ex-ample, using excision to transfer the question to a closed pair .X;†/ and thenusing the constructions which were used to compare orientations in Proposi-tion 2.17. This observation allows us to write ƒ.˛; ˇ/, omitting the z.

Because G is simple, the bundle P admits a product connection B0 forwhich ' is parallel. We add a singular term in the standard way, to obtaina connection B' with a codimension-two singularity; the monodromy of thisconnection lies in the one-parameter subgroup generated by ˆ. We let �' de-note the corresponding point in B.Y;K;ˆ/. This point is neither irreducibleor non-degenerate, so we cannot defineƒ.�' ; ˛/ as above because the operatorQA will not be Fredholm as it stands. To remedy this, we we can regard QAas acting weighted Sobolev space, on which this operator is Fredholm. Thatis, we choose a connection A in Cloc from B' to B˛ and define ƒ.�' ; ˛/ asthe set of orientations of the determinant line of the operator QA acting in thetopologies

QA W e��t LL2m;A0 ! e��t LL2m�1;A0

on the infinite cylinder. Here � is a small positive constant, smaller than thesmallest positive eigenvalue of the extended Hessians � and ˛.

Now that we have a basepoint �' , we can define a 2-element set

ƒ.˛/ D ƒ.�' ; ˛/:

78

We could equally well defineƒ.˛/ asƒ.˛; �'/ (with the same weighted Sobolevspaces), because of the composition law and the fact thatƒ.˛; ˛/ is canonicallytrivial.

With this understood, the composition law for the orientation lines givesus a map

ƒ.˛/ �ƒ.ˇ/! ƒ.˛; ˇ/:

If ˛ and ˇ are now critical points and ŒA� is a solution of the equations be-longing to the (regular) moduli space Mz.˛; ˇ/, then ƒ.˛; ˇ/ is isomorphic tothe set of orientations of the moduli space at ŒA�. Using the above composi-tion law, we can turn this round and say that an orientation of Mz.˛; ˇ/ at ŒA�determines an isomorphism ƒ.˛/! ƒ.ˇ/.

In particular, we can consider the case that grz.˛; ˇ/ D 1. In this case, themoduli space of unparametrized trajectories MMz.˛; ˇ/ is a finite set of points,and Mz.˛; ˇ/ is a finite set of copies of R, acted on by the translations ofthe cylinder. Thus Mz.˛; ˇ/ is canonically oriented. To be quite specific, if �tdenotes the translation .s; y/ 7! .sCt; y/ of R�Y , we make R act onMz.˛; ˇ/

by ŒA� 7! ��t ŒA�, and we use this to give each orbit of R an orientation. Foreach ŒA� in Mz.˛; ˇ/, we therefore obtain an isomorphism

�ŒA� W ƒ.˛/! ƒ.ˇ/: (82)

3.7 Floer homology

We can now define the Floer homology groups. The situation is that we havea compact, connected, oriented 3-manifold Y with an oriented knot or linkK � Y , a choice of simple, simply-connected Lie group G and a ˆ in thefundamental Weyl chamber with �.ˆ/ < 1. A Riemannian metric g� with anorbifold singularity along K is given. We continue to suppose that the non-integrality condition (Definition 3.2) holds and that a perturbation � 2 P

is chosen so as to satisfy Hypothesis 3.19. We also need to suppose that ˆsatisfies the monotone condition, Definition 2.4.

For a 2-element setƒ D f�; �0gwe use Zƒ to mean the infinite cyclic groupobtained from the rank-2 abelian group Z�˚ Z�0 by imposing the condition� D ��0. Thus a choice of element of ƒ determines a generator for Zƒ. Wedefine C�.Y;K;ˆ/ to be the free abelian group

C�.Y;K;ˆ/ DMˇ2C�

Zƒ.ˇ/; (83)

79

If grz.˛; ˇ/ D 1 and Œ MA� denotes the R-orbit of some ŒA� in Mz.˛; ˇ/, thenfrom (82) above we obtain an isomorphism

�Œ MA� W Zƒ.˛/! Zƒ.ˇ/:

Combining all of these, we define

@ W C�.Y;K;ˆ/! C�.Y;K;ˆ/

by@ D

X.˛;ˇ;z/

XŒ MA�

�Œ MA� (84)

where the first sum runs over all triples with grz.˛; ˇ/ D 1.That the above sum is finite depends on the monotonicity condition. The

point is that for any pair .˛; ˇ/, there will be infinitely many homotopy classesof paths z with grz.˛; ˇ/ D 1 (as long as K is non-empty). Thus the first sumin the definition of @ has an infinite range. The monotone condition, however,ensures that only finitely many of the 1-dimensional moduli spaces Mz.˛; ˇ/

will be non-empty: this is the statement of Corollary 3.25.Based as usual on a gluing theorem and consideration of the compactifica-

tion of moduli spaces MMz.˛; / with grz.˛; / D 2, one shows that @B@ D 0. Itis important here that the moduli spaces of broken trajectories MMCz .˛; / arecompact when grz.˛; / D 2, as follows from Proposition 3.22.

Definition 3.26. When the non-integrality and transversality assumptions ofHypothesis 3.19 holds and ˆ satisfies the monotone condition, we defineI�.Y;K;ˆ/ to be the homology of the complex .C�.Y;K;ˆ/; @/.

Since grz.˛; ˇ/ taken modulo 2 is independent of the path z, we can regardI�.Y;K;ˆ/ as having an affine grading by Z=2. For particular choices of Gand ˆ, the greatest common divisor of grz.ˇ; ˇ/, taken over all closed paths,may be a proper multiple of 2, in which case I�.Y;K;ˆ/ has an affine Z=.2d/-grading for d > 1. For example, if G D SU.N / and ˆ has just two distincteigenvalues, then the homology is graded by Z=.2N /.

Rather than being left as a relative (i.e. affine) grading, the mod 2 gradingcan be made canonical. For a critical point ˛, the grading of ˛ mod 2 canbe defined as the mod 2 reduction of grz.�

' ; ˛/, where �' is the reducibleconfiguration constructed earlier and z is any homotopy class of paths. Theresult is independent of the choices made.

80

3.8 Cobordisms and invariance

The Floer group I�.Y;K;ˆ/ depends only on .Y;K/ as a smooth orientedpair and on the choice of ˆ: it is independent of the remaining choices made.These choices include the choice of Riemannian metric and the perturbation� : changing either of these may change the set of critical points that form thegenerators of the complex. More subtly, the choice of cut-off function involvedin the construction of the base connection B' may effect the 2-element setƒ.˛/ used in fixing signs. As in Floer’s original approach, the independenceof the Floer groups on these choices can be seen as a consequence of a moregeneral property, namely the fact that a cobordism between pairs gives rise toa homomorphism on Floer homology.

To say this more precisely, let .Y0; K0/ and .Y1; K1/ be two pairs. Bya cobordism between them we will mean a connected, oriented manifold-with-boundary, W , containing a properly embedded oriented surface-with-boundary, S , together with an orientation-preserving diffeomorphism of pairs

r W . NY0; NK0/q .Y1; K1/! .@W; @S/:

If .W; S/ and .W 0S 0/ are two cobordisms between the same pairs, then anisomorphism between them means a diffeomorphism between the underlyingmanifolds commuting with r . Isomorphism classes of cobordisms can be com-posed in the obvious way, and in this manner we obtain a category, whose ob-jects are the pairs .Y;K/ and whose morphisms are the isomorphism classes ofcobordisms. If .W1; S1/ is a cobordism from .Y0; K0/ to .Y1; K1/ and .W2; S2/is a cobordism from .Y1; K1/ to .Y2; K2/, we denote by

.W; S/ D .W2 BW1; S2 B S1/ (85)

the composite cobordism from .Y0; K0/ to .Y2; K2/.We adopt from [24] the appropriate definition of a homology orientation

for a cobordism W from Y0 to Y1: a homology orientation oW is a choice oforientation for the line

ƒmaxH 1.W IR/˝ƒmaxIC.W /˝ƒmaxH 1.Y1IR/

where IC.W / is a maximal positive-definite subspace for the non-degeneratequadratic pairing on the image of H 2.W; @W IR/ in H 2.W IR/. (Note thatthe links Ki and the 2-dimensional cobordism S are not involved here, andare omitted from our notation.) This definition can be made to look less ar-bitrary by regarding this as an orientation for the determinant line of the op-erator �d� ˚ dC on the cylindrical-end manifold obtained from W , acting

81

on weighted Sobolev spaces with a consistent choice of weights. There is acomposition law for homology orientations: if W D W2 B W1 and homologyorientations oWi are given, we can construct a homology orientation oW2 BoW2for W . This is most easily seen from the second description of what a homol-ogy orientation is. We thus have a modified category in which the morphismsare cobordisms of pairs, .W; S/, equipped with homology orientations, up toisomorphism.

Let .W; S/ be a cobordism from .Y0; K0/ to .Y1; K1/. Suppose that eachYi is equipped with a Riemannian metric and that perturbations �i are chosensatisfying Hypothesis 3.19. We continue to suppose also that ˆ satisfies themonotone condition. Let base connections B'ii be chosen for each. In thiscase, we have Floer homology groups I�.Yi ; Ki ; ˆ/, for i D 0; 1, which de-pend a priori on the choices made. Let us temporarily denote this collection ofchoices (of metric, perturbation and base connection) by �i , and so write thegroups as

I�.Yi ; Ki ; ˆ/�i ; i D 0; 1:

The fact that cobordisms give rise to maps can be stated as follows.

Proposition 3.27. Suppose that ˆ satisfies the monotone condition. For i D0; 1, let .Yi ; Ki / be pairs as above, and suppose that Hypothesis 3.19 holds forboth. Let �i be choices of Riemannian metric, connectionB' and perturbation asabove. Let .W; S/ be a cobordism from .Y0; K0/ to .Y1; K1/, and let a homologyorientation oW for the cobordism W be given. Then .W; S; oW / gives rise to ahomomorphism

I�.W; S;ˆ; oW / W I�.Y0; K0; ˆ/�0 ! I�.Y1; K1; ˆ/�1 (86)

which depends only on the isomorphism class of the cobordism with its homologyorientation. Furthermore, composition of cobordisms becomes composition ofmaps and the trivial product cobordism gives the identity map.

Remark. The choice of homology orientation oW affects only the overall signof the map I�.W; S;ˆ; oW /, and affects it non-trivially only if the dimensionof G is odd: cf. Proposition 2.17.

In particular, by taking W to be a cylinder and setting

.Y0; K0/ D .Y1; K1/ D .Y;K/;

we see that the Floer group I�.Y;K/� is independent of the auxiliary choices� , up to canonical isomorphism. Usually, we omit mention of oW and � from

82

our notation (just as we have already silently omitted r), and we simply write

I�.W; S;ˆ/ W I�.Y0; K0; ˆ/! I�.Y1; K1; ˆ/

with the unstated understanding that the identifications r (when needed) areimplied and that oW is needed to fix the overall sign of this map if the dimen-sion of G is odd.

The proof of Proposition 3.27 follows standard lines, and can be modelled(for example) on the arguments from [24]. We content ourselves here with someremarks about the construction of the maps I�.W; S;ˆ/.

For i D 0; 1, let ˇi be a critical point in B.Yi ; Ki ; ˆ/. On the pair .W; S/,let us consider aG-bundle P equipped with a section ' ofOP along S and cor-responding connection A with singularity along S , subject to the constraintthat the restrictions of A to the two ends should define singular connectionsbelonging the gauge-equivalences classes of ˇ0 and ˇ1. There is an obviousnotion of a continuous family of such data, .Pt ; 't ; At / parametrized by anyspace T , and we can therefore consider the set of deformation-classes of suchdata. We will refer to such an equivalence class as a path from ˇ0 to ˇ1 alongthe cobordism .W; S/. In the case of a cylindrical cobordism, such a path isthe same as a homotopy class of paths from ˇ0 to ˇ1 in B.Y;K;ˆ/. If S hasany closed components, then different paths along .W; S/ may also be distin-guished by having different monopole charges on the closed components. If.W; S/ is a composite cobordism, as in (85), and if z1 and z2 are paths along.W1; S1/ and .W2; S2/ from ˇ0 to ˇ1 and from ˇ1 to ˇ2 respectively, then thereis a well-defined composite path along .W; S/, obtained by choosing any iden-tification of the two bundles on Y1 respecting the sections 'i and the connec-tions.

Remark. There is a small point to take note of here. By assumption, the criticalpoint ˇ1, like all critical points, is irreducible and has stabilizer Z.G/. Whenforming the composite path by identifying the two bundles along Y1, there istherefore a Z.G/’s worth of choice in how the identification is made. Despitethis choice, the composite path is well-defined, because the automorphisms ofthe connection on Y1 extend to the 4-manifolds.

Let W C be the manifold obtained by attaching cylindrical ends to the twoboundary components of W , and let this manifold be given a Riemannianmetric g�W which is a product metric on each of the two cylindrical pieces. LetSC � W C be obtained similarly from S .

Let critical points ˇi in B.Yi ; Ki ; ˆ/ be given for i D 0; 1, and let z bea path along .W; S/ from ˇ0 to ˇ1. Let .PW ; 'W ; AW / be a representative

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for z, and extend this data to the cylindrical ends by pull-back. Imitating thedefinition of Cz.˛; ˇ/ from (80), we define a configuration space of singularconnections Cz.W; S;ˆIˇ0; ˇ1/ as the space of all A differing from AW by aterm belonging to LL2m, and we write Bz.W; S;ˆIˇ0; ˇ1/ for the correspondingquotient space.

Let �0 and �1 be the chosen holonomy perturbations on Y0 and Y1 respec-tively. We perturb the 4-dimensional equations on W C by adding a term sup-ported on the cylindrical ends: this term will be a t -dependent holonomy per-turbation �W equal to �i on the two ends. In more detail, in a collar Œ0; 1/�Y0of one of the boundary components Y0 � W , the perturbed equations take theform

FCA C ˇ.t/OU0.A/C ˇ0.t/ OV0.A/ D 0

where, as in (78), OV0 is the perturbing term defined by �0 2 P and OU0 is definedby a choice of an auxiliary element of P . The cut-off function ˇ is supportedin the interior of the interval, while ˇ0 is equal to 1 near t D 0 and equal to 0near t D 1. (This choice of perturbation follows [24, Section 24].) We write

Mz.W; S;ˆIˇ0; ˇ1/ � Bz.W; S;ˆIˇ0; ˇ1/

for the moduli space of solutions to the perturbed equations on W C. Forgeneric choice of auxiliary perturbation OUi on the two collars, the moduli spaceis cut out transversely by the equations and (under our standing assumptionsof Hypothesis 3.19) is a smooth manifold. A choice of homology orienta-tion oW and an element of ƒ.ˇ0/ and ƒ.ˇ1/ determines an orientation of themoduli space. As in the closed case, if G is even-dimensional, then oW is notneeded. The map (86) is defined in the usual way by counting with sign thepoints of all zero-dimensional moduli spaces Mz.W; S;ˆIˇ0; ˇ1/. As in thedefinition of the boundary map @, the monotonicity condition ensures thatthis is a finite sum, because for fixed ˇ0 and ˇ1, the dimension of the modulispace corresponding to a path z along .W; S/ is an affine-linear function of thetopological energy.

3.9 Local coefficients

There is a standard way in which the construction of Floer homology groupscan be generalized, by introducing a local system of coefficients, �, on the con-figuration space (in this case, the configuration space B.Y;K;ˆ/ of singularconnections modulo gauge transformations on the 3-manifold). Thus we sup-pose that for each point ˇ in the configuration space we have an abelian group

84

�ˇ and for each homotopy class of paths z from ˛ to ˇ and isomorphism �zfrom �˛ to �ˇ satisfying the usual composition law. If we make the same as-sumptions as before (the conditions of Hypothesis 3.19 and the monotonicitycondition, Definition 2.4), then we can modify the definition of the chain groupC�.Y;K;ˆ/ by setting

C�.Y;K;ˆI�/ DMˇ2C�

Zƒ.ˇ/˝ �ˇ

and taking the boundary map to be

@ DX.˛;ˇ;z/

XŒ MA�

�Œ MA�˝ �z : (87)

The homology of this complex, I�.Y;K;ˆI�/ is the Floer homology with co-efficients �.

If we are given two pairs, .Y0; K0/ and .Y1; K1/ with local systems �0 and�1, and if .W; S/ is a cobordism between the pairs, then we have a naturalnotion of morphism,�, of local systems along .W; S/: such a� assigns to eachpath z from ˇ0 to ˇ1 along .W; S/ (in the sense of the previous subsection) ahomomorphism

�z W �0ˇ0! �1ˇ1 (88)

respecting the composition maps with paths in B.Yi ; Ki ; ˆ/ on the two sides.(See [24], for example.) Using such a morphism �, we can adapt the definitionof the map I�.W; S;ˆ/ in an obvious way to obtain a homomorphism

I�.W; S;ˆI�/ W I�.Y0; K0; ˆI�0/! I�.Y1; K1; ˆI�

1/:

To give an example, we begin with a standard local system �S1

on thecircle S1, regarded as R=Z, defined as follows. We write R for the ring of finiteLaurent series with integer coefficients in a variable t . This is the group ringZŒZ�, and we can regard it as lying inside the group ring ZŒR�: the ring offormal finite series X

x2R

axtx :

For each � in R, we have an R-submodule t�R � ZŒR� generated by the ele-ment t�: this is the R-module of all finite series of the formX

x2�CZ

axtx :

85

As � varies in R=Z these form a local system of R-modules, �S1

over S1: themap �S

1

z corresponding to a path z is given by multiplication by t�1��0 if zlifts to a path in R from �0 to �1. If we are now given a circle-valued function

� W B.Y;K;ˆ/! S1 D R=Z

then we can pull back the standard local system �S1

to obtain a local system

�� D ��.�S1

/

on B.Y;K;ˆ/.This construction can be applied using a class of naturally-occurring circle-

valued functions on the configuration space of singular connections. Thesefunctions can be defined, roughly speaking, by taking the holonomy of a con-nection B along a longitudinal curve close to a component of the link K andapplying a character of Gˆ. To say this more precisely, we choose a framing ofthe link K � Y , so has to have well-defined coordinates on the tubular neigh-borhood, up to isotopy, identifying the neighborhood with D2 �K. Supposefirst that K has just one component, and for each sufficiently small � > 0, letT� be the torus obtained as the product of the circle of radius � in D2 withknot K. Use the coordinates to identify T� with S1 �K. If B is a connectionin C.Y;K;ˆ/, then by restricting to T� we obtain in this way a sequence of G-connections on S1 � K; and the definition of the space LL2m in which we workguarantees that these have a well-defined limit, up to gauge transformation,which is a flat connection B0 on S1 � K. The holonomy of B0 along a curvebelonging to the S1 factor is exp.'/, and the holonomy along the longitudinalcurve belonging to the K factor lies in the commutant. Choose a character

s W Gˆ ! U.1/

and letw W gˆ ! R

be the corresponding weight, so that s.exp.x// D e2�iw.x/. We can apply sto the holonomy of B0 along the longitudinal curve to obtain a well-definedelement of U.1/, depending only on the gauge-equivalence class of B. Thus weobtain from s a function

�s W B.Y;K;ˆ/! U.1/ D R=Z (89)

by applying s to the holonomy along the longitudinal curve. In this way weobtain a local system ��s by pull-back.

86

The choice of framing of K is essentially immaterial. The set of framingsis an affine copy of Z; and if we change the chosen framing of K by 1, then �sis changed by the addition of the constant w.ˆ/ mod Z. The correspondinglocal systems are canonically isomorphic, via multiplication by tw.ˆ/.

IfK has more than one component, we can apply this construction to eachone, perhaps using different characters s, and form the product. Alternatively,one could define a local system over a ring of Laurent series in a number ofvariables ti , one for each component of K.

In the above construction, the reason for taking such a specifically-definedfunction �s, rather than a general circle-valued function belonging to the samehomotopy class, is that the naturality inherent in the construction leads to aFloer homology group that is a topological invariant of the pair, rather than agroup that is an invariant only up to isomorphism. The point is that if we havea cobordism of pairs, .W; S/, with a chosen framing of a tubular neighborhoodof S (or at least of the components of S having non-empty boundary), then weobtain a natural morphism � between the corresponding local systems asso-ciated to the framed knots at the two ends. The map �z corresponding to apath z along .W; S/ can be defined as follows. Fix data .P; '; A/ on W corre-sponding to z. For each small positive �, we have a copy of S1 � S in W , asthe boundary of the �-neighborhood of S in its framed tubular neighborhoodD2 � S , and we therefore obtain connections A� on S1 � S . The limit of theseconnections is a connection A0 on S1 � S whose curvature 2-form has the S1

direction in its kernel. ThusA0 gives aGˆ-connection on S1�S ; and applyingthe character s we obtain a U.1/-connection s.A0/ on S1�S . For any p in S1,we have a parallel copy of S as fpg � S , and the map �z can then be definedas multiplication by t� , where

� Di

2�

Zfpg�S

Fs.A0/: (90)

Because the curvature 2-form of s.A0/ annihilates the circle directions, we seethat we could have taken any section of S1 � S instead of the constant sectionfpg � S , and the above integral would be unchanged. So in the end, the map�z is independent of the choice of framing of S .

Local systems can also be made use of to define Floer groups in the casethatˆ does not satisfy the monotone condition. Whenˆ is not monotone, thesum (87) which defines the boundary operator may have infinitely many non-zero terms; but the sum can still be made sense of if each �˛ is a topologicalgroup and the local system is such that the series converges. A typical instance

87

of such a construction replaces the ring R of finite Laurent series which weused above by the ring of Laurent series that are infinite in one direction.

3.10 Non-simple groups

We have been considering instanton Floer homology in the case that G is asimple group. When discussing instanton moduli spaces, we saw in section 2.10how the definitions are readily adapted to the case which of a non-simple groupsuch as the unitary group. We now carry this over to the Floer homology set-ting. We again suppose that G has a simply-connected commutator subgroup.We write Z.G/ for its center and NZ.G/ for G=ŒG;G�. Unlike the case in whichG itself is simply-connected, it is no longer the case that a G-bundle P ! Y

must be trivial: its isomorphism type is determined by the NZ.G/-bundle d.P /,or equivalently by the characteristic class c D c.P / in H 2.Y IL.G// of (51).

To preserve the functoriality of the Floer homology groups, we need toadopt the alternative viewpoint for the configuration space and gauge groupwhich we mentioned briefly in subsection 2.10. We fix NZ.G/-bundle ı ! Y

with an isomorphism q W d.P / ! ı, and we fix a connection ‚ in ı. Asbefore, we let ‚' denote the corresponding singular connection in ı (equation(53)), and we construct a space C.Y;K;ˆ/ı of singular connections, B, withthe constraint that d.B/ D q�.‚'/. The gauge group G .Y;K;ˆ/ consists ofgauge transformations g of class LL2mC1 with d.g/ D 1 and we have a quotientspace B.Y;K;ˆ/ı .

The construction of the Floer groups then proceeds as before, with straight-forward modifications of the same type as we dealt with in section 2.10. Wedeal with some of these modifications in the next few paragraphs.

The Chern-Simons functional. The appropriate Chern-Simons functional onC.Y;K;ˆ/ı in the present setting is the one which ignores the central com-ponent of the connection: it can be defined by the same formula (66) as be-fore, if we understand that the inner products in (66) are defined using thesemi-definite Killing form. Critical points of the unperturbed Chern-Simonsfunctional on C.Y;K;ˆ/ı are singular connections B such that the inducedconnection NB with structure group G=Z.G/ in the adjoint bundle is flat. Theformal gradient flow lines of this functional correspond to connections A intemporal gauge on the cylinder with the property that NA is anti-self-dual.

The non-integral condition. The most important change involves the non-integrality condition, Definition 3.2, which we used to rule out reducible crit-

88

ical points and which formed part of our standing Hypothesis 3.19. In thecase thatG is not simple, the corresponding condition can be read off from the4-dimensional version, Proposition 2.19:

Definition 3.28. Let the components ofK again beK1,. . . ,Kr . For non-simplegroups G, we will say that the bundle P on .Y;K;ˆ/ satisfies the non-integralcondition if, in the notation of section 2.10, the expression

w˛.c.P //C

rXjD1

. Nw˛ B �j /. N /P:D:ŒKj �

is a non-integral cohomology class for every choice of fundamental weight w˛and Weyl group elements �1; : : : ; �r .

The simplest example in which this non-integrality holds is the case corre-sponding to Corollary 2.20, in which G is the unitary group U.N/ and all thecomponents of Ki are null-homologous: in this case, the non-integral condi-tion is equivalent to saying that the pairing of c1.P / with some integral ho-mology class in Y is coprime to N .

As previously, we need to suppose that this non-integrality condition holdsand that further, as in Hypothesis 3.19, the stabilizer in G .Y;K;ˆ/ı of everycritical point is exactly Z.G/ \ ŒG;G�, rather than some larger finite group (acondition which is automatic in the non-integral case if G D U.N/). Underthese conditions, and when ˆ satisfies the monotone condition (57), we willarrive at a Floer homology group

I�.Y;K;ˆ/ı

depending on the choice of bundle NZ.G/-bundle ı.

Holonomy perturbations. The definition of holonomy perturbations does notneed any changes in the case of more general G. The basic ingredient is still achoice of function

h W Gr ! R

invariant under the diagonal action of G, acting by the adjoint representationon each factor. Holonomy perturbations still separate points in the quotientspace B.Y;K;ˆ/ı . Note that the choice of connection‚ is involved in the con-struction, because we are taking the holonomy of aG-connectionB in the bun-dle P which satisfies d.B/ D ‚' . If we chose h so that it was pulled back from.G=Z.G//r , then the choice of‚ would again become irrelevant; but functionsh of this sort are not a large enough class, as they do not allow our holonomyperturbations to separate points and tangent vectors in B.Y;K;ˆ/ı .

89

Orientations. In the case of a simple group G, we defined a 2-element setƒ.˛; ˇ/ for a pair of configurations ˛ and ˇ; and we then defined ƒ.˛/ asbeing ƒ.�' ; ˛/, where �' was a specially chosen connection. The importantfeatures of our choice of �' were first that �' was reducible and second that,although the construction depended on details such as a choice of cut-off func-tion, any two choices differed by a small isotopy, so that an essentially uniquepath connects any two choices.

When P is not simple and d.P / is non-trivial, we do not have a distin-guished gauge-equivalence class of trivial connections in P from which to con-struct �' , but we can instead proceed as we did in section 2.10. We fix again ahomomorphism e W NZ.G/! T which is right-inverse to d (see (60)). As in the4-dimensional case, we obtain aG-connection e.‚/ on a bundle isomorphic toP , with d.e.‚// D ‚. After adding the singular term alongK, we obtain a dis-tinguished gauge-equivalence class of connections, �' , in B.Y;K;ˆ/ı . Once e

is fixed, this gauge-equivalence class depends only on the details of how the sin-gular term is constructed, through the choice of cut-off function for example.This puts us in a position to define ƒ.˛/ as we did before.

Cobordisms. Let .W; S/ now be a cobordism of oriented pairs, and write itstwo boundary components as .Yi ; Ki / for i D 0; 1, so that

@.W; S/ D . NY0; NK0/q . NY1; NK1/:

(In the slightly more categorical language that we used earlier in section 3.8,we are supposing here that the identification map r is the identity.) Let ıW bea NZ.G/-bundle on W , and let write

ıi D ıW jYi ; i D 0; 1:

Fix G-bundles P0 and P1 on Y0 and Y1 with isomorphisms qi W d.Pi / ! ıi .Let ‚0 and ‚1 be chosen connections in ı0 and ı1.

We wish to show how the data .W; S; ıW / (together with a homologyorientation of W ) gives rise to a homomorphism from I�.Y0; K0; ˆ/ı0 toI�.Y1; K1; ˆ/ı1 . The first step is to extend our previous notion of a “pathalong .W; S/” between critical points ˇ0 and ˇ1 belonging the configurationspaces B.Yi ; Ki ; ˆ/ıi for i D 0; 1. To do this, we let ˇi be represented by sin-gular connections Bi on Pi ! Yi and we define a path z to be defined by dataconsisting of:

� a bundle P ! W ;

90

� an isomorphism qW W d.P /! ıW ;

� a reduction of structure group defined by a section ' ofOP along S ; and

� an isomorphism R from .P0 q P1/ to P j@W , respecting the reductionof structure group along Ki and such that d.R/ fits into the obviouscommutative diagram involving the other maps on the NZ.G/-bundles –a condition which appears as

qW B d.R/ D .q0 q q1/:

For any path z in this sense, we can construct a moduli space, generalizingour earlier Mz.W; S;ˆIˇ0; ˇ1/. To do this, we use the data P , Pi , Bi andR to construct a bundle PC on the cylindrical-end manifold W C, togetherwith a connection on the two cylindrical ends (obtained by pulling back theBi ). We extend this connection arbitrarily to a connection AW on the wholeof W C, with a singularity along SC, and we write ‚'W for d.AW /. We canthen define a configuration space Cz.W; S;ˆIˇ0; ˇ1/ıW and quotient spaceCz.W; S;ˆIˇ0; ˇ1/ıW using singular connections A satisfying d.A/ D ‚

'W

and with A � AW of class LL2m. Introducing perturbations as before, we arriveat a moduli space Mz.W; S;ˆIˇ0; ˇ1/ıW . The task of orienting this modulispace is the same as the case of a simple group G, with slight modificationsdrawn from section 2.10, so that when homology orientation of W is giventogether with elements of ƒ.ˇ0/ and ƒ.ˇ1/, the moduli space is canonicallyoriented.

We summarize the situation with a proposition, generalizing Proposi-tion 3.27 to the case of non-simple groups:

Proposition 3.29. Suppose that ˆ satisfies the monotone condition (57). Let.W; S/ be a cobordism with boundary the two pairs .Yi ; Ki / as above, let ıW be aNZ.G/-bundle and let ıi be its restriction to Yi . Suppose that the non-integrality

condition Definition 3.28 holds at both ends and Hypothesis 3.19 holds, so thatthe Floer groups I�.Yi ; Ki ; ˆ/ıi are defined. Then, after choosing a homologyorientation oW , there is a well-defined homomorphism

I.W; S;ˆ/ıW W I�.Y0; K0; ˆ/ı0 ! I�.Y1; K1; ˆ/ı1 : (91)

These homomorphisms satisfy the natural composition law when a cobordism.W; S/ is decomposed as the union of two pieces and ıW is restricted to eachpiece.

91

There is an important point about the naturality of this construction thatis worth spelling out in more detail. Rather than taking ıi to be the restrictionof ıW to the boundary component Yi , we could have taken the NZ.G/-bundlesı0 and ı1 to have been given in advance, in which case it is more natural toregard the necessary data on the cobordism W as consisting of

� a NZ.G/-bundle ıW ! W ; and

� a pair of isomorphisms Qr D . Qr0; Qr1/ from ıW jYi to ıi , i D 0; 1.

In this setting, the induced map between the two instanton homology groupsI�.Yi ; Ki ; ˆ/ıi does depend on the choice of isomorphisms Qri . The followingcorollary of Proposition 3.29 makes essentially the same point:

Corollary 3.30. Given .Y;K/ and a NZ.G/-bundle ı satisfying as usual the condi-tions of Hypothesis 3.19, the group of components of the bundle automorphismsof ı ! Y acts on the the instanton homology group I�.Y;K;ˆ/ı .

Proof. Given an automorphism g of ı, consider the cobordism .W; S/ thatis the cylinder Œ0; 1� � .Y;K/ and the bundle ıW which is the pull-back. Wecan identify ıW with ı by using the identity map at the boundary componentf0g � Y and the map g at the other boundary component f1g � Y . From thedata .W; S/ and ıW with these identifications, .r0; r1/ D .1; g/, we obtain ahomomorphism from I�.Y;K;ˆ/ı to itself.

Some of the automorphisms of ı act trivially on the Floer homology:

Proposition 3.31. Suppose that G is constructed from G1 D ŒG;G� as in (50)and that Condition 2.18 holds. If G1 is SU.N /, then suppose additionally that Gis U.N/ and that e is standard. (See Proposition 2.21.)

Then an automorphism g W ı ! ı of the NZ.G/-bundle ı ! Y acts triviallyon I�.Y;K;ˆ/ı if g has the form d.f / for some Z.G/-valued automorphismf W P ! P of the corresponding bundle P !Y.

Proof. TakeW to be the cylinder Œ0; 1��Y , and let ıW be as in the proof of theprevious corollary. If g D d.f /, then we can describe ıW as ı1 ˝ d.�/, whereı1 is the pull-back bundle Œ0; 1� � ı and � ! W is a Z.G/ bundle equippedwith a trivialization at each boundary component of W . That is, we take � tobe W �Z.G/, with the trivialization 1 at f0g � Y and f at f1g � Y .

We are therefore left to compare two maps from I�.Y;K;ˆ/ı to itself: thefirst is the identity map, arising from the product cobordismW with ı1; and thesecond is the map obtained fromW with ı1˝d.�/. This is essentially the same

92

situation as the construction of the map �� in (61). In particular, the modulispaces that are involved in defining the two maps are identical, and the onlyquestion is whether the zero-dimensional moduli spaces arise with the samesign. Proposition 2.21 tells us that �� preserves orientation in all cases exceptSU.N / and E6. For the SU.N / case, �� still preserves orientation because thecobordismW has even intersection form on its relative homology. For the caseof E6, an examination of the proof of Proposition 2.21 shows that orientationdepends on a term �. Ne/ Y �. Ne/; so again, the even intersection form ensuresthat �� is orientation-preserving.

The bundle automorphisms of ı are the maps Y ! NZ.G/ and the group ofcomponents is H 1.Y I�1. NZ.G///. The above Proposition tells us that the im-age ofH 1.Y I�1.Z.G// acts trivially. Under the hypotheses of Condition 2.18,we have a short exact sequence

�1.Z.G//! �1. NZ.G//! Z.G1/

in which the first two groups are free abelian and the first map is multiplica-tion by p. From the corresponding long exact sequence in cohomology, welearn that the largest group that may act effectively on I�.Y;K;ˆ/ı via thisconstruction is isomorphic to the subgroup

H � H 1.Y IZ.G1//

consisting of the elements with integer lifts:

H D im�H 1.Y I�1. NZ.G///! H 1.Y IZ.G1//

�:

As explained in section 2.10 (where we treated the 4-dimensional case),we can also regard the automorphisms of ı as defining, rather directly, auto-morphisms of the configuration space B.Y;K;ˆ/ı . Were it the case that theholonomy perturbations could be chosen to be invariant under the action ofH while still achieving the necessary transversality, then we would have a moredirect way of understanding the action of H on the instanton homology: theaction on the set of critical points would give rise to an action of H on thechain complex C�.Y;K;ˆ/ by chain maps. Although perturbations cannot al-ways be chosen so as to realize the action in this way, the following situationdoes arise in some cases:

Proposition 3.32. Suppose that a subgroup H 0 � H acts freely on the set ofcritical points for the unperturbed Chern-Simons functional CS in B.Y;K;ˆ/ı .

93

Then a holonomy perturbation � can be found, as in Proposition 3.12, that isinvariant under H 0, such that the critical point set C� is non-degenerate, theaction of H 0 on C� is still free, and the moduli spaces Mz.˛; ˇ/ are all regular.

Proof. A cylinder function arising from a collection of loops and a maph W Gr ! R will be invariant under the action of H 0 on B.Y;K;ˆ/ı providedthat h is invariant under an associated action of H 0 on Gr (given by multi-plications by central elements). This gives us a means to construct invariantperturbations, and the statements about the critical point set are straightfor-ward. For the moduli spaces Mz.˛; ˇ/, we note that, by unique continuation,once the action on C� is known to be free, it must also be free on the subsetof B.Y;K;ˆ/ consisting of all points lying on gradient trajectories betweencritical points. Once the action is known to be free here, the transversalityarguments go through without change.

4 Classical knots and variants

4.1 Summing with a 3-torus

Take G to be the group U.N/, let Y be any closed, oriented 3-manifold andK � Y an oriented knot or link. Let y0 be a base-point in Y n K, and let anoriented frame in Ty0Y be chosen. Using the base-point and frame, we canform the connected sum Y#T 3 in a manner that makes the result unique towithin a canonical isotopy class of diffeomorphisms. The knot or link K nowbecomes a knot or link in the connected sum. Any topological invariant thatwe define for the pair .Y#T 3; K/ becomes an invariant of the original pair.Y;K/ together with its framed basepoint.

Regard the 3-torus as a product, S1 � T 2, and let ı1 ! T 3 be a U.1/bundle with c1.ı1/ Poincare dual to S1 � fpointg. Extend ı1 trivially to theconnected sum Y#T 3, and call the resulting U.1/-bundle ı. If P is a U.N/bundle on Y#T 3 whose determinant is ı, then P satisfies the non-integral con-dition, Definition 3.28, for any choice of ˆ in the positive Weyl chamber. (Seethe remarks immediately following that definition.) The remaining conditionin Hypothesis 3.19 (that the stabilizer of each critical point is just Z.G /) isautomatically satisfied in the case of U.N/ once the non-integrality conditionholds. There is therefore a well-defined Floer homology group (in the notationof section 3.7),

I�.Y#T 3; K;ˆ/ı

94

for any choice ˆ in the positive Weyl chamber which satisfies the monotonecondition.

Let us consider a particular choice of ˆ in this context, namely

ˆ D diag.i=2; 0; : : : ; 0/:

This ˆ satisfies the monotone condition, and its orbit in g D u.N / is a copyof CPN�1. The group element exp.2�ˆ/ has order 2: it is

diag.�1; 1; : : : ; 1/: (92)

We introduce a notation for the corresponding instanton homology group:

Definition 4.1. We write FIN� .Y;K/ for the instanton homology group

I�.Y#T 3; K;ˆ/ı

in the case that G D U.N/, with ı and ˆ as above. In the case that Y isS3, we simply write FIN� .K/; and in the case that N D 2, we write FI�.Y;K/or FI�.K/. The group FIN� .Y;K/ has an affine grading by Z=.2N / (see theremark following Definition 3.26).

Remark. Ifˆ is changed by the addition of a central element of u.N /, then theresulting instanton homology is essentially unchanged. Thus, we could equallywell have taken ˆ to be the element

ˆ0 D idiag.1=2; 0; : : : ; 0/ � .i=.2N //diag.1; 1; : : : ; 1/ (93)

in su.N /, so that

exp.2�ˆ0/ D e��i=Ndiag.�1; 1; : : : ; 1/ 2 SU.N /: (94)

To examine what comes of this definition, let us begin by looking atFIN� .;/, i.e. the case of the the empty link in S3. In other words, we are lookingat I�.T 3/ı1 . Let a, b and c be standard generators for the fundamental groupof T 3, with a and b generating the fundamental group of the T 2 factor inT 3 D T 2 �S1. Let p 2 T 2 be a point not lying on the a or b curves, and letDbe a small disk around p. We can take the line bundle ı1 to be pulled back fromT 2, with a trivialization on the complement ofD�S1. Let‚1 be a connectionin the line bundle ı1 ! T 3 that is also pulled back from T 2. We can take the‚1 to respect the trivialization of ı1 on the complement D � S1, so that thecurvature of ‚1 is a 2-form supported in that neighborhood. If A 2 B.T 3/ı1

95

is a critical point for the Chern-Simons functional, then the restriction of A toT 2 nD is a flat SU.N / connection whose holonomy around @D is the centralelement e2�i=N in SU.N /. In this way, the critical points correspond to conju-gacy classes of triples fh.a/; h.b/; h.c/g in SU.N / (the holonomies around thethree generators) satisfying

Œh.a/; h.b/� D e2�i=N

Œh.a/; h.c/� D 1

Œh.b/; h.c/� D 1:

There are N different solutions to these conditions (see also [19]): the elementh.c/ can be any of theN elements of the center of SU.N /, and up to the actionof SU.N /, we must have

h.a/ D �

26666641 0 0 � � � 0

0 � 0 � � � 0

0 0 �2 � � � 0: : : 0

0 0 0 0 �N�1

3777775

h.b/ D �

26666640 0 0 � � � 1

1 0 0 � � � 0

0 1 0 � � � 0: : : 0

0 0 0 1 0

3777775

h.c/ D �k

26666641 0 0 � � � 0

0 1 0 � � � 0

0 0 1 � � � 0: : : 0

0 0 0 0 1

3777775 ; k 2 f0; 1; : : : ; N � 1g:

(95)

Here, � D e2�i=N , and � is 1 if N is odd and an N th root of �1 if N is even.Thus the Chern-Simons functional has exactly N distinct critical points in

B.T 3/ı1 . These critical points are irreducible (as they must be, on account ofthe coprime condition); and it is shown in [19] that these N critical points arenon-degenerate. We can now describe the critical points in the case of a general.Y;K/.

96

Proposition 4.2. For any oriented pair .Y;K/, the set of critical points of theChern-Simons functional on B.Y#T 3; K;ˆ/ı consists of N disjoint copies ofthe space of representations

� W �1.Y nK/! SU.N / (96)

satisfying the condition that, for each oriented meridian m of the knot or link K,the element �.m/ is conjugate to (94).

Proof. Rather than consider B.Y#T 3; K;ˆ/ı as in the statement of the propo-sition, we may alternatively consider the isomorphic space

B.Y#T 3; K;ˆ0/ı

with the alternative element ˆ0 2 su.N / from (93). As in the discussion ofT 3 above, the critical points in B.Y#T 3; K;ˆ0/ı correspond to flat SU.N /connections on the complement of K q .D � S1/ in Y#T 3, such that theholonomy around @D is e2�i=N and �.m/ is in the conjugacy class of (94) forall oriented meridians m. The fundamental group of the complement in thisconnected sum is a free product, so the result follows from the fact that thereare N such flat connections on the T 3 summand, all of which are irreducible:see the discussion surrounding (4) in the introduction.

Corollary 4.3. In the case of the 3-sphere, and an oriented classical knot orlink K � S3, the set of critical points of the Chern-Simons functional onB.S3#T 3; K;ˆ/ı consists of N disjoint copies of the space of representations

� W �1.S3nK/! U.N/ (97)

satisfying the condition that, for each oriented meridian m of the knot or link K,the element �.m/ is conjugate to (92).

Proof. In the case of S3, the meridians generate the first homology of the com-plement of the link, and the space of U.N/ representations that appears hereis therefore identical to the space of SU.N / representations in the propositionabove.

As a special case, we have:

Corollary 4.4. For the unknot K in S3, the set of critical points

C � B.S3#T 3; K;ˆ/ı

consists of N disjoint copies CPN�1. Furthermore, the Chern-Simons functionis Morse-Bott: at points of C, the kernel of the Hessian is equal to the tangentspace to C.

97

Proof. Since the fundamental group of the knot complement is Z, a homomor-phism � as in the previous corollary is determined by the image of a meridianm in the conjugacy class (92). This conjugacy class in U.N/ can be identi-fied with CPN�1 by sending an element to its .�1/-eigenspace. The kernel ofthe Hessian can be computed from Lemma 3.13, to establish the Morse-Bottproperty.

We introduce a notation for the space of representations that appears inthe previous proposition:

Definition 4.5. We write

R.Y;K;ˆ0/ � Hom.�1.Y nK/;SU.N //

for the space of homomorphisms � such that, for all meridians m, the element�.m/ is conjugate to (94).

Recall from section 3.10 that the set of critical points is acted on by thegroup H , which in our present case is the subgroup

H � H 1.Y#T 3IZ=N/

consisting of elements with integer lifts. Let

H 0 Š Z=N

H 0 � H

be the subgroup of H 1.T 3IZ=N/ consisting of elements which are non-zeroonly on the generator c in �1.T 3/ and are zero on the generators a and b.

Lemma 4.6. The action of this copy, H 0, of Z=N on the set of critical points,C, in B.Y#T 3; K;ˆ/ı is to cyclically permute the N copies of R.Y;K;ˆ0/ thatcomprise C according to the description in Proposition 4.2.

Proof. From our description of C, it follows that it sufficient to check thelemma for the case of the N critical points in B.T 3/ı1 (i.e. the case that Yis S3 and K is empty). These critical points are described in (95), and theaction of H 0 Š Z=N is to multiply h.c/ by the N th roots of unity.

We can compare the values of the Chern-Simons functional on the Ncopies of R.Y;K;ˆ0/ in C. For example, in the case of the unknot, becauseCPN�1 is connected, the functional is constant on each copy and we can com-pare the N values. The general case is the next lemma.

98

Lemma 4.7. Let s 2 H 0 be the generator that evaluates to 1 on the generator cin T 3. Then for any ˛ D ŒA� in B.Y#T 3; K;ˆ/ı we have

CS.s.˛// � CS.˛/ D �16�2.N � 1/ (98)

modulo the periods of the Chern-Simons functional.

Proof. The calculation reduces to a calculation for a connection ŒA� on T 3

and its image under s. Pull back the U.N/ bundle P1 (with d.P1/ D ı1) to thecylinder Œ0; 1� � T 3. Identify the two ends to form S1 � T 3, gluing the bundleP1 using an automorphism f of P1 with d.f / D u. Let P ! S1 � T 3 be theresulting U.N/ bundle. We have

c1.P / D PD�ŒS1 � c�C Œa � b�

�modulo multiples ofN . The change in the Chern-Simons functional is half the“topological energy” E.P / on S1 � T 3, so

CS.s.˛// � CS.˛/ D �8�2p1.gP /ŒS1 � T 3�;

from (56). Using the relation (55), we obtain

CS.s.˛// � CS.˛/ D �8�2c1.P /2ŒS1 � T 3� (99)

modulo periods, and the final result follows from the above formula for c1.P /,which gives

c1.P /2ŒS1 � T 3� D 2 .mod N/:

We can also consider the relative grading for the pair of points ˛ and s.˛/in B.Y#T 3; K;ˆ/ı along a suitable chosen path z:

grz.s.˛/; ˛/ 2 Z:

Note that this relative grading is easy to interpret unambiguously, even whenthe Hessian at ˛ has kernel, essentially because the Hessians at ˛ and s.˛/ areisomorphic operators.

Lemma 4.8. Let z be a path in B.Y#T 3; K;ˆ/ı along which a single-valued liftof the Chern-Simons functional satisfies (66). Then along this path we have

grz.s.˛/; ˛/ D �4.N � 1/:

99

Proof. Concatenating z with its image under the maps s, s2, . . . , sN�1, weobtain a closed loop, along which the total energy E is �32�2.N � 1/N . Fromthe monotone relationship between the dimension and energy, we have that thespectral flow along the closed loop is�4.N�1/N . The spectral flow along eachpart is therefore �4.N � 1/, because each part makes an equal contribution.

According to Proposition 3.32, we can choose a holonomy perturbation� which is invariant under H 0 while still making the critical point set non-degenerate and the moduli spaces regular. If follows that we have an actionof H 0 on FIN� .Y;K/ resulting from this geometric action on the configura-tion space. (Without using an invariant perturbation, the action can still bedefined – using cobordisms – by the procedure described in subsection 3.10.)Since the grading in FIN� .Y;K/ is only defined modulo 2N , we can interpretthe last lemma above as saying that the action of s gives an automorphism ofFIN� .Y;K/ of degree 4:

s� W FINj .Y;K/! FINjC4.Y;K/:

(The subscript is to be interpreted mod 2N .)Whereas the construction using cobordisms only gives us an action on the

homology, the geometric action on the configuration space gives us an actionon the chain complex. So, rather than consider the action of H 0 Š Z=N on theinstanton homology group, we can instead consider dividing B.Y#T 3; K;ˆ/ıby the action of H 0 and then taking the Morse homology. We can interpretProposition 3.32 as telling us that � can be chosen so that the Morse construc-tion works appropriately on B.Y#T 3; K;ˆ/ı=H 0. The relative grading on theMorse complex is defined mod 4 if N is even, and mod 2 if N is odd.

Definition 4.9. We define NFIN� .Y;K/ to be the homology of the quotient chaincomplex NC�.Y#T 3; K;ˆ/ı : the quotient of C�.Y#T 3; K;ˆ/ı by the action ofZ=N . We write NFIN� .K/ in the case that Y is the 3-sphere.

We can calculate this group in the case of the unknot.

Proposition 4.10. For the unknot K in S3, we have

NFIN� .S3; K/ Š H�.CPN�1IZ/

Š ZN ;

FI N� .S3; K/ Š H�.CPN�1 q � � � qCPN�1IZ/; (N copies)

Š ZN2

:

100

Proof. The group NFIN� .S3; K/ is the homology of NC�.Y#T 3; K;ˆ/ı , and this

chain complex has generators corresponding to the points of C�=H0, for suit-

able holonomy perturbation � . Before perturbation, C� consists of N copiesof CPN�1 and H 0 is a copy of Z=N which permutes the N copies cyclically.So C=H 0 is a single copy of CPN�1.

Choose a holonomy perturbation �1 which is invariant under H 0 and issuch that the corresponding function f1 on B.S3#T 3; K/ı has the propertythat f1jC is a standard Morse function with even-index critical points on eachcopy of CPN�1. Then set �� D ��1 and take � a small, positive quantity.Because the Chern-Simons functional is Morse-Bott, an application of the im-plicit function theorem and the compactness theorem for critical points showsthat, for � sufficiently small, the perturbed critical set C��=H

0 consists of ex-actlyN critical points. As � goes to zero, these converge to theN critical pointsof f1jCPN�1 , and the relative grading of the points in C�� along paths in theneighborhood of C is equal to the relative Morse grading of the correspondingcritical points of f1jCPN�1 . It follows that, for this perturbation, the complexhas N generators, all of which are in the same grading mod 2.

In the case N D 2, the invariant NFI�.K/ of classical knots appears to re-semble Khovanov homology in the simplest cases. As mentioned in the intro-duction, it is natural to ask whether we have

NFI�.K/ Š Kh.K/

for the .2; p/, .3; 4/ and .3; 5/ torus knots, for example.

4.2 Bridge number and representation varieties

For a knot K in S3, the knot group (i.e. the fundamental group of the knotcomplement) is generated by the conjugacy class of the meridian. If we choosemeridional elements m1; : : : ; mk which generate the knot group, then a homo-morphism � as in (97) from the knot group to U.N/ is entirely determined byk elements

Ai D �.mi /

in the conjugacy class of the reflection (92): or equivalently, k points inCPN�1. The �1-eigenspaces of the reflections Ai will span at most a k-planein CN . It follows that � is conjugate in U.N/ to a representation whose imagelies in U.k/ � U.N/.

101

In this sense, the problem of describing the space of representations � sta-bilizes at N D k. For larger N , the homomorphisms � to U.N/ are ob-tained from the homomorphisms to U.k/ by conjugating by elements of thelarger unitary group. An upper bound for the number k of meridians thatare needed to generate the knot group is the bridge number of the knot. Sofor a k-bridge knot, the problem of computing the set of critical points C inB.S3#T 3; K;ˆı/ for the group U.N/ can be reduced to the correspondingproblem for U.k/ (though the critical point sets are not the same).

The simplest example after the unknot is the trefoil, a 2-bridge knot. Thegroup is generated by a pair of meridians, and for N D 2 a representation �is therefore determined by a pair of points in CP1 Š S2. The relation be-tween the generators in the knot group implies that these two points in S2

either coincide or make an angle 2�=3. The set of all such representations forN D 2 is therefore parametrized by one copy of S2 and one copy of SO.3/.When N is larger, we have essentially the same classification: a representationis determined by a pair of points in CPN�1, and either these coincide, or theylie at angle 2�=3 from each other along the unique CP1 that contains themboth. These two components are a copy of CPN�1 itself and a copy of theunit sphere bundle in TCPN�1 respectively. The authors conjecture that forthe trefoil, NFI

N� .K/ is isomorphic to the direct sum of the homologies of these

two components of the critical set of the unperturbed functional.For a general knot K, the critical set C, after perturbation, determines the

set of generators of the complex that computes FIN� .K/. It would be inter-esting to know whether there is any sort of stabilization that occurs for thedifferentials in the complex, as N increases. The situation is reminiscent of thelarge-N stabilization for Khovanov-Rozansky homology that is discussed in[10, 36].

4.3 A reduced variant

In the construction of FIN� .Y;K/, the important feature of the manifold T 3

with which we formed the connected sum was that, for a suitable choice ofU.N/ bundle P1 ! T 3, the corresponding set of critical points C.T 3; P1/ wasjust a finite set of reducibles (a single orbit of the finite group H 0 Š Z=N ).Rather than T 3, we can consider the pair .S1 � S2; L/, where L � S1 � S2 isthe .N C 1/-component link

L D S1 � fp0; : : : pN g:

102

Let P0 ! S1 � S2 be the trivial SU.N / bundle and let ˆ0 2 su.N / be theelement (93). In the space of singular connections B.S1 �S2; L;ˆ0/, consideragain the set of critical points:

C.S1 � S2; L;ˆ0/ � B.S1 � S2; L;ˆ0/:

The pair .S1�S2; L/with this choice ofˆ0 fits the hypotheses of Corollary 2.8,and it follows that the critical set consists only of irreducible flat connections.

Lemma 4.11. The critical set C.S1 � S2; L;ˆ0/ consits of exactly N non-degenerate, irreducible points. These form a single orbit of the group H D

H 1.S1 � S2IZ=N/.

Proof. The critical set parametrizes conjugacy classes of homomorphisms

� W �1.S1� S2 n L/! SU.N /

such that � of each meridian of L is conjugate to exp.ˆ0/. The fundamentalgroup is a product, with a Z factor coming from the S1. The lemma willfollow if we can show that there is only a single, irreducible, conjugacy class ofhomomorphisms

� W �1.S2n fp1; : : : pNC1g/! SU.N /

such that � sends the linking circle of each pj into the conjugacy class ofexp.ˆ0/. The classification of such homomorphisms � can be most easilyachieved by using the correspondence with stable parabolic bundles on theRiemann sphere with .N C 1/ marked points. In this instance, the relevantparabolic bundles are holomorphic bundles E ! CP1 of rank N and degree0, equipped with a distinguished line Li in the fiber Epi for each i . The ap-propriate stability condition for such a parabolic bundle .E; fLig/ is that, forevery proper holomorphic subbundle F � E, we should have

#f i j Li � Fpi g C 2 deg.F / � rank.F /:

The only solution to these constraints is to take E to be the trivial bundleO˝CN and to take Li to be Opi ˝Li , where the lines Li define N C 1 pointsin general position in CPN�1. There is therefore a single homomorphism �

satisfying the given conditions.

Now let .Y;K/ be an arbitrary pair, and let k0 be a basepoint onK. Choosea framing of K at k0, and use this framing data to form the connected sum ofpairs

. OY ; OK/ D .Y;K/#.S1 � S2; L/;

103

connecting the component of K containing k0 to the component S1 � fp0g ofL. We define a reduced version of the framed instanton homology by setting

RIN� .Y;K/ D I�. OY ; OK;ˆ0/:

Like FIN� .Y;K/, this group has an affine grading by Z=.2N /.The definition is such that, for the case that Y is S3 and K is the unknot,

the pair . OY ; OK/ is simply .S1 � S2; L/. The lemma above thus tells us that,in this case, the complex that computes RIN� has N generators. The relativegrading of these generators is even, and we therefore have

RIN� .S3; unknot/ D ZN :

The set of critical points in B. OY ; OK;ˆ0/ for a general .Y;K/ can be de-scribed by a version of Proposition 4.2. Let us again write R.Y;K;ˆ0/ for thespace of homomorphisms described in Definition 4.5. As a basepoint for thefundamental group �1.Y nK/ let us choose the push-off the the chosen pointk0 2 K using the framing. We then have a preferred meridian,m0, in �1.Y nK/linking K near this basepoint, and hence an evaluation map taking values inthe conjugacy class C.expˆ0/ of the element exp.ˆ0/:

ev W R.Y;K;ˆ0/! C.expˆ0/

� 7! �.m0/:(100)

The counterpart to Proposition 4.2 is then:

Proposition 4.12. For any oriented pair .Y;K/, the set of critical points of theChern-Simons functional on B. OY ; OK;ˆ0/ consists of N copies of the fiber of theevaluation map (100).

Thus, for example, in the case of the trefoil, the set of critical points consistsof N points and N copies of the sphere S2N�3.

As in the case of FIN� , it is possible to pass to the quotient of the con-figuration space by the action of the cyclic group H 0. The result is a variant,NRIN� .Y;K/, which is isomorphic to Z in the case of the unknot. In the quotient

space B. OY ; OK;ˆ0/=H 0, the set of critical points consists of just one copy of thefiber of the evaluation map ev above.

Rasmussen [38] has observed that the reduced Khovanov homology coin-cides with Heegaard-Floer knot homology group 1HFK.K/, defined by Ozsvathand Szabo in [32] and by Rasmussen in [37], for many knots, but not for the.4; 5/ torus knot. It would therefore be interesting to have some data compar-ing the reduced group NRI�.K; k0/ with 1HFK.K/.

104

4.4 Longitudinal surgery

Another variant briefly mentioned in the introduction is to begin with a null-homologous knot K in an arbitrary Y , and form the pair .YK ; K0/, where YKis the 3-manifold obtained by 0-surgery (longitudinal surgery), and K0 is thecore of the solid torus used in the surgery. The knot K0 represents a primitiveelement in the first homology of the manifold YK . We can apply our basicconstruction to this pair (without taking a further connected sum), with G DSU.N / as usual. To avoid reducibles, we can take ˆ to have just two distincteigenvalues with coprime multiplicity (see Corollary 2.7). In particular, wemay again take the ˆ0 given in equation (93). We make a definition to coverthis case:

Definition 4.13. For a null-homologous knot K in a 3-manifold Y , we defineLIN� .Y;K/ to be the result of applying the standard construction I� to theoriented pair .YK ; K0/, with G D SU.N / and ˆ given by (93).

For N D 2, we just write LI�.Y;K/, and if Y D S3 we drop the Y fromour notation.

The complement ofK0 in YK is homeomorphic to the original complementofK in Y , and the meridian ofK0 corresponds to the longitude ofK. Thus wesee that the set of critical points of the Chern-Simons function in the configu-ration space for .YK ; K0/ can be identified with the space of conjugacy classesof homomorphisms

� W �1.Y nK/! SU.N / (101)

satisfying the constraint that � maps the longitude of K to an element conju-gate to exp.2�iˆ/. In the case of the unknot in S3, the group is Z and thelongitude represents the identity element, so the set of critical points is empty.For the unknot therefore, the group LIN� .K/ is zero. The same applies to an“unknot” in any Y , i.e. a knot that bounds a disk.

In part because of the results of [23] and [22], it is natural to conjecture thatLI�.Y;K/ is zero only ifK is an unknot. An examination of the representationvariety, and a comparison with Casson’s work [1], suggests that the Euler char-acteristic of LI�.K/ should be 2�00K.1/, where �K is the symmetrized Alexan-der polynomial. In the case of a torus knot K, the representation variety ofhomomorphisms � satisfying the longitudinal constraint

�.longitude/ ��i 0

0 �i

�

105

consists of exactly 2�00K.1/ points, and it would follow that

LI�.K/ D Z2�00K.1/

for all torus knots. (For the .2; p/ torus knot with p > 3, it would followfrom this that LI�.K/ is not isomorphic to Eftekhary’s longitude variant ofHeegaard Floer homology [12]; because for the .2; p/ knot, the quantity�00K.1/grows quadratically with p, while the rank of Eftekhary’s knot invariant growslinearly.)

5 Filtrations and genus bounds

In this section, we take up a theme from the introduction and explore how alower bound for the slice genus of a knot can be obtained from (a variant of)instanton Floer homology. In doing so, we will see that the formal outline ofthe construction can be made to resemble closely the work of Rasmussen in[35], while the underlying mechanism of the proof draws on [18].

From this point on, we will work exclusively with the group G D SU.2/and the element ˆ D i diag.1=4;�1=4/ in the fundamental alcove (this is theonly balanced case for the group SU.2/). We will write FI�.K/ for the framedinstanton homology of a classical knot, as described in Definition 4.1, andFI�.Y;K/when the knot is in a 3-manifold other than S3. To keep the notationto a minimum, we will treat only classical knots to begin with, though littlechanges when we generalize to knots in other 3-manifolds.

5.1 Laurent series and local coefficients

Let K � S3 be an oriented knot, and let us write

B.K/ D B.S3#T 3; K;ˆ/ı :

for the configuration space that is used in defining the framed instanton Floerhomology group, Definition 4.1. As in section 3.9, we will consider the Floerhomology ofK coming from a system of local coefficients on the configurationspace B.K/. Specifically, we let � W B ! U.1/ D R=Z be a circle-valuedfunction arising from the standard character Gˆ ! U.1/, using the holonomyconstruction of (89); and we let � D �� be the corresponding local systempulled back from R=Z, as described in section 3.9. Thus � is a local system offree modules of rank 1 over the ring

R D ZŒZ�

D ZŒt�1; t �:

106

Thus for each knot K, we have a finitely-generated R-module

FI�.KI�/:

We should recall at this point that the construction of the local system ��

really depends on a choice of framing n for the knot K, but that the localsystems arising from different choices of framings are all canonically identified(see section 3.9 again). So we should regard � as the common identification ofa collection of local systems ��n as n runs through all framings.

A cobordism of pairs, .W; S/, from .S3; K0/ to .S3; K1/ gives rise to ahomomorphism of the corresponding Floer groups, by the recipe described at(90). (The way we have set it up, a framing of the normal bundle to S is neededin order to define the map, but the resulting map is independent of the choicesmade.) We will abbreviate our notation for the map and write

W;S W FI�.K0I�/! FI�.K1I�/:

A homology-orientation of W is needed to fix the sign of the map W;S .From the argument of Proposition 4.10 we obtain:

Proposition 5.1. For the unknot U in S3, the Floer group FI�.U I�/ is a freeR-module of rank 4.

The definition of the local coefficient system � and the maps induced by acobordism are related to the monopole number. To understand this relation-ship, consider in general two different cobordisms of pairs, .W; S/ and .W 0; S 0/from .S3; K0/ to .S3; K1/, (with both S and S 0 being embedded surfaces, notimmersed). Let ˇ0 and ˇ1 be critical points in B.K0/ and B.K1/ respectively,and let z and z0 be paths from ˇ0 to ˇ1 along .W; S/ and .W 0; S 0/ respectively.Corresponding to z and z0, the local system gives us maps of the form (88),which in this case means we have

�z; �z0 W t�.ˇ0/R! t�.ˇ1/R:

Both of these maps are multiplication by a certain (real) power of t :

�z D t�.z/

�z0 D t�.z0/:

We can express the difference between �.z/ and �.z0/ in topological terms. Thesurfaces S and S 0 are not closed, so there is not a well-defined monopole num-ber l for the classes z and z0; but there is a well-defined relative monopolenumber: we can write

d.z � z0/ D .k; l/

107

where k is the relative instanton number and l is the relative monopole number(both are integer-valued). There is a also a “relative” self-intersection numberS2 � .S 0/2 (the self-intersection number of the union of S and �S 0 in W [.�W 0/. In these terms, we have

�.z/ � �.z0/ D �l C .1=4/.S2 � .S 0/2/:

This is essentially the formula (17) of [18], which expresses the curvature inte-gral which defines � in terms of the topological data. Our � corresponds to ��in [18]. If we fix a reference cobordism .W�; S�/ and a path z� along it fromˇ0 to ˇ1, then the contribution involving ˇ0 and ˇ1 to the map W;S can bewritten (in the style of definition (84)) asX

z

XŒ MA�

�Œ MA��z DXz

XŒ MA�

�Œ MA� t�lC.1=4/.S2�.S�/

2/C�z� : (102)

5.2 Immersed surfaces and canonical isomorphisms

A cobordism of pairs from .Y0; K0/ to .Y1; K1/, as considered so far, con-sists of a 4-dimensional cobordism W and an embedded surface S with @S DK1 � K0. It is convenient to follow [18] and consider also immersed surfacesS . We will always consider only smoothly immersed surfaces f W S # W

with normal crossings (transverse double-points), all of which should be in theinterior of W . As is common, we often omit mention of f and confuse S withits image in W . By blowing up W at each of the double points (forming a con-nected sum with copies of NCP

2) and taking the proper transform of S (cf.[18],

we have a canonical procedure for replacing any such immersed cobordism.W; S/ with an embedded version, . QW ; QS/. We then define a map

W;S W FI�.K0I�/! FI�.K1I�/

corresponding to the immersed cobordism by declaring it to be equal to themap obtained from its resolution:

W;S WD QW ; QS :

Now suppose that f0 W S ! W and f1 W S ! W are two immersions in Wwith normal crossings, and suppose that they are homotopic as maps relativeto the boundary. Then the image S1 of f1 can be obtained from S0 D f .S0/

by a sequence of standard moves, each of which is either

108

(0) an ambient isotopy of the image of the immersed surface in W ,

(1) a twist move introducing a positive double point,

(2) a twist move introducing a negative double point,

(3) a finger move introducing two double points of opposite sign,

or the inverse of one of these [14]. To analyze the relation between the maps W;S0 and W;S1 , we must therefore analyze the effect of each of these typesof elementary changes to an immersed surface. This was carried out in [18]for the case of closed surfaces in a closed 4-manifold, and the same argumentswork as well in the relative case, leading to the following result.

Proposition 5.2. Let S be obtained from S 0 by one of the three basic moves (1)–(3). Then the maps

W;S 0 W FI�.K0; �/! FI�.K1; �/

W;S W FI�.K0; �/! FI�.K1; �/

are related by, respectively,

(1) W;S D .t�1 � t / W;S 0 ,

(2) W;S D W;S 0 (no change), and

(3) W;S D .t�1 � t / W;S 0 (the same as case (1)).

Remark. The three cases of this proposition can be summarized by saying thatthe map W;S acquires a factor of .t�1� t / for every positive double point thatis introduced.

Proof. As indicated above, this is essentially Proposition 3.1 of [18]. In thatproposition, the monopole number l contributes to the power of t in the co-efficients of the map W;S , according to the formula (102). The other contri-bution to the exponent is the self-intersection number of the proper transformof the immersed surface S , which changes by �4 when S acquires a positivedouble point and is unaffected by negative double-points.

Let us now return to situation where we have two homotopic maps fi WS ! W with images Si , i D 0; 1. If we form the ring R0 by inverting theelement .t�1 � t / in R, so

R0 D RŒ.t�1 � t /�1�;

109

and if we write

0W;S D W;S ˝ 1 W FI�.K0I�/˝R R0 ! FI�.K1I�/˝R R0

then the proposition tells us:

Corollary 5.3. If S0 and S1 are the images of homotopic immersions as above,then the maps

0W;Si W FI�.K0I�/˝R R0 ! FI�.K1I�/˝R R0

differ by multiplication by a unit. More specifically, if �.Si / is the number ofpositive double-points in Si , then we have

0W;S1 D .t�1� t /�.S1/��.S0/ 0W;S0 :

Corollary 5.4. For any two knots K0 and K1, we have

FI�.K0I�/˝R R0 Š FI�.K1I�/˝R R0

Proof. In the cylindrical cobordism W D Œ0; 1� � S3, let S be any immersedannulus from K0 to K1, and let NS be any annulus from K1 to K0. The con-catenation of these immersed annular cobordisms, in either order, give annularimmersed cobordisms from K0 to K0 and from K1 to K1. These compositeannuli are each homotopic to a trivial product annulus; so the composite maps

0W;S ı 0

W; NS

and 0W; NSı 0W;S

are both the identity map, and it follows that 0W;S and 0W; NS

are isomor-phisms.

Extending this line of thought a little further, we see that the groupFI�.KI�/ ˝R R0 is not just independent of K up to isomorphism, but upto canonical isomorphism. That is, if K0 and K1 are any two knots, we maychoose any immersed annulus S from K0 to K1 in the 4-dimensional productcobordisms W D Œ0; 1� � S3 and construct the isomorphism

.t�1 � t /��.S/ 0W;S W FI�.K0I�/˝R R0 ! FI�.K1I�/˝R R0:

This isomorphism is independent of the choice of annulus S . In particular, forany knot K, the R0-module FI�.KI�/˝R R0 is canonically isomorphic to theR0-module arising from the unknot. From Proposition 5.1 we therefore obtain:

Corollary 5.5. For any knot K in S3, there is a canonical isomorphism

‰ W .R0/4 ! FI�.KI�/˝R R0:

110

5.3 Filtrations and double-point bounds

The inclusion R ! R0 gives us a canonical copy of R4 in .R0/4, the imageof FI�.U I�/ in FI�.U I�/ ˝R R0 for the unknot U . We define an increasingfiltration of FI�.KI�/˝R R0,

� � � � F �1.K/ � F 0.K/ � F 1.K/ � � � �

by first settingF 0.K/ D ‰.R4/ � FI�.KI�/˝R R0;

where ‰ is the canonical isomorphism of Corollary 5.5, and then defining

F i .K/ D .t�1 � t /�iF 0.K/: (103)

Although F 0.U / is the image of FI�.U I�/ in the tensor product for the caseof the unknot, this does not hold for a general knot. We make the followingdefinition, modelled on similar constructions in [35, 31, 26].

Definition 5.6. For any knotK in S3 we define %.K/ to be the smallest integer isuch that F �i .K/ is contained in the image of FI�.KI�/ in FI�.KI�/˝R R0.

To see that the definition makes sense, choose an immersed annular cobor-dism from the unknot U to K, and let � be the number of positive doublepoints in this annulus. As R-submodules of FI�.KI�/ ˝R R0, we have, fromthe definitions,

F �� .K/ D ‰..t�1 � t /�R4/

D 0W;S�FI�.U I�/

�� im

�FI�.KI�/! FI�.KI�/˝R R0/

where the last inclusion holds because passing to the ring R0 commutes withthe maps W;S and 0W;S induced by the cobordism. From this observationand the definition of %.K/, we obtain

%.K/ � �:

Since the annulus S was arbitrary, we have:

Theorem 5.7. Let K be a knot in S3 and let D be an immersed disk with normalcrossings in the 4-ball, with boundary K. Then the number of positive doublepoints in D is at least %.K/.

111

5.4 Algebraic knots

The lower bound for the number of double points in an immersed disk, givenby Theorem 5.7, is sharp for the case of an algebraic knot (a knot arising asthe link of a singularity in a complex plane curve, such as a torus knot). Thereason this is so comes down to the same mechanisms that were involved in[20, 21] and [18], where singular instantons were used to obtain bounds onunknotting numbers and slice genus.

To explain this, we recall some background from [20, 18]. Let .X;†/ be aclosed pair, with † connected for simplicity, and let P ! X be a U.2/ bundle.Suppose that c1.P / satisfies the non-integrality condition, that

12c1.P /˙

14Œ†�

is not an integer class, for either choice of sign. Denoting by k the instantonnumber of P , we have for any choice of monopole number l a moduli spaceMk;l.X;†/ı , which we label by k l and ı, where ı is the line bundle detP . If theformal dimension of this moduli space is zero, then there an integer invariant

qık;l.X;†/ 2 Z:

(A homology orientation is needed as usual to fix the sign.) In [18], theseinteger invariants are combined into a Laurent series (with only finitely manynon-zero terms):

Rı.X;†/.t/ D 2�g.†/X

.k;l/WdimMk;lD0

t�lqık;l.X;†/:

The normalizing factor 2�g was convenient in [18] but is not significant here.The definition of qı

k;l.X;†/ and Rı.t/ is extended to the case of immersed

surfaces with normal crossings by blowing up.Suppose now that .X;†/ decomposes along a 3-manifold Y , meeting †

transversely in a knot K. Suppose also that the restriction of P to .Y;†/

satisfies the non-integrality condition. Then there is a gluing formula whichexpresses the Laurent series Rı.X;†/.t/ as a pairing between a cohomologyclass and a homology class in the Floer group

I�.Y;K;P I�/:

The coefficient system � is the one we have been using, and keeps track of thepower of t .

112

Figure 1: A closed pair .X;†/ separated by .W; S/, with S immersed.

Suppose now that we have a cobordism .W; S/ (with S immersed perhaps)from .Y0; K0/ to .Y1; K1/, giving us a map

W I�.Y0; K0; P I�/! I�.Y1; K1; P I�/:

Suppose we wish to show that the image of is not contained in the imageof multiplication by .t�1 � t /. From the functorial properties and the gluingformulae, it will be sufficient if we can find a closed pair .X;†/ (together witha U.2/ bundle P ) such that:

� .X;†;P / contains .W; S; P / as a separating subset, as indicated in thefigure; and

� the Laurent series Rı.X;†/.t/ does not vanish at t D 1.

Summarizing this discussion, we therefore have:

Proposition 5.8. Suppose .X;†/ is a pair (with † perhaps immersed) such that,for some ı, the finite Laurent series Rı.X;†/.t/ is non-vanishing at t D 1. Sup-pose that .X;†/ has a decomposition as shown, and suppose:

� Y0 Š Y1 Š S3#T 3;

� W the 4-dimensional product cobordism;

� c1.ı/ is dual to a standard circle in T 3;

� K0, K1 arise from classical knots in S3, with K0 the unknot;

113

� the surface S arises from an immersed annulus in Œ0; 1� � S3 with � doublepositive points.

Then %.K1/ D � and the bound of Theorem 5.7 is sharp for K1.

Consider now the 4-torus T 4 as a complex surface, containing an algebraiccurve C with a unibranch singularity at a point p. Let B1 be a small ballaround p so that the curve C meets @B1 in a knot K1. In a C1 manner, wecan alter C in the interior of B1 to obtain an immersed surface QC , so thatthe part of QC that is in the interior of B1 is isotopic to a complex-analyticimmersed disk with � positive double-points. Let B0 � B1 be a smaller 4-ball,meeting QC in a standard embedded disk, so that the part of QC that lies betweenB0 and B1 is an immersed annulus with � double points. Let T be a real 2-torus in T 4 disjoint from C [ B, and let ı be a line bundle with c1.ı/ŒT � D 1.Let Y1 Š S3#T 3 be obtained as an internal connected sum of @B1 with theboundary of a tubular neighborhood of T . Similarly, let Y0 be obtained asthe internal connected sum of @B0 with the boundary of a smaller tubularneighborhood of T . Then the pair .T 4; QC/ has a decomposition as shown inthe diagram, satisfying the itemized conditions of the theorem above.

To show that %.K1/ D � for this algebraic knot K1, we therefore need onlyshow that the corresponding Laurent series Rı.T 4; QC/.t/ is non-zero at t D 1.Because of the results of [18], this is equivalent to showing that Rı.T 4; †/.1/is non-zero, where † is a smooth algebraic curve (embedded in the abeliansurface). Using the results of [20] however, we can calculate this Laurent series.We are free to arrange that † has odd genus, that c1.ı/Œ†� is zero, and thatc1.ı/

2 D 0. The terms in the series come from the invariants qık;l

with

2k C l � 12.g � 1/ D 0;

and from [20] we learn that

qık;l D

(2gqı0.T

4/; k D 0 and l D .g � 1/=2,

0; otherwise,

where qı0.T4/ is the Donaldson invariant of T 4, which is 2. The Laurent series

is therefore a non-zero multiple of a certain power of t , and in particular isnon-zero at t D 1, as required.

5.5 The involution on the configuration space

As we have defined it, the group FI�.KI�/ is a free R-module of rank 4 in thecase that K is the unknot. The four generators come from the two 2-spheres

114

that make up the set of critical points of the unperturbed Chern-Simons func-tional on B.K/. However, there is an involution on B.K/, interchanging thesetwo copies: this is the action of the cyclic group H 0 of order 2. Recall that,with Z coefficients, we defined NFI�.K/ (in Definition 4.9, where we dealt withSU.N / for arbitrary N ) by passing to the quotient B.K/=H 0 and taking theMorse theory in this quotient.

Because the local coefficient system � on B.K/ is actually pulled back fromthe quotient B.K/=H 0, we can adapt this construction to define an R-module

NFI�.KI�/ (104)

for any knot K. For a suitable choice of perturbation, the complex that com-putes NFI�.KI�/ is the quotient of the complex that computes FI�.KI�/ by aninvolution that acts freely on the generators. In the case of the unknot, thisFloer group would be R2 instead of R4. Little else in our discussion wouldneed to be changed.

5.6 Genus bounds

Letf W S ! W

f 0 W S 0 ! W

be two immersions with transverse self-intersections, having as boundary thesame knots K0 � Y0 and K1 � Y1. We have seen that if S D S 0 andf ' f 0 relative to the boundary, then the two resulting maps FI�.K0I�/ !FI�.K1I�/ differ only by factors of .t�1 � t / (Proposition 5.2 and Corol-lary 5.3). Another situation to consider is the case that S 0 is obtained fromS by adding a handle: forming an internal connected sum with a 2-torus con-tained in a ball in W .

The effect of adding a handle in this way was examined for the case ofclosed pairs .X;†/ in [18]. In our present context, the relevant calculation isthe following. Let W be the 4-dimensional product cobordism Œ0; 1� � S3, andlet S1 � W be a cobordism from the unknot to the unknot and having genus1. This gives rise to a homomorphism

1 W FI�.U I�/! FI�.U I�/:

If we pass to the “bar” version of the Floer groups (104), then we have also amap

N 1 W NFI�.U I�/! NFI�.U I�/:

115

The group NFI�.U I�/ is a free R-module of rank 2, with generators in differentdegrees mod 4. We can therefore identify it with R ˚ R with an ambiguityconsisting of multiplication by units on each summand. The map N 1 must beoff-diagonal in this basis, because its degree is 2 mod 4; so we have a map

N 1 W R˚R! R˚R

of the form

N 1 D

�0 p.t/

q.t/ 0

�(105)

for certain Laurent polynomials p.t/ and q.t/, well-defined up to units. From[18] we know the effect of adding two handles to a surface, and from that wededuce the relation

p.t/q.t/ D 4.t � 2C t�1/;

or in other wordsN 21 D 4.t � 2C t

�1/

D 4t�1.t � 1/2:(106)

as an endomorphism of NFI�.U I�/.At this point, because of the factor of 4 in the above formula, we shall

pass to rational coefficients rather than integer coefficients: without change ofnotation, let us redefine R as QŒt�1; t �. We again define R0 by inverting .1� t /and .1C t / in R.

Suppose now that we have an embedded cobordism S of genus g from theunknot U to K, inside Œ0; 1� � S3. This gives rise to a map

N S W NFI�.U I�/! NFI�.KI�/

and similarly

N 0S WNFI�.U I�/˝R R0 ! NFI�.KI�/˝R R0:

The surface S is homotopic to an immersed cobordism SC which is a compos-ite of two parts: the first part is g copies of the standard genus-1 cobordism S1fromU toU ; and the second part is an immersed annulus. Let � be the numberof positive double points in the immersed annulus. From Corollary 5.3 and thedefinition of the canonical isomorphism ‰, we obtain

N 0S D ‰ ı .N 01/

g

116

where N 01 is the map defined by (105). This provides a constraint on the genusg: it must be that ‰ ı . N 01/

g carries R ˚ R into the image of NFI�.U I�/ inNFI�.U I�/ ˝R R0. This constraint gives us a lower bound for g, just as we

obtained a lower bound %.K/ for the number of double points previously. Foralgebraic knots again, the bound will be sharp.

It is not inconceivable that, by working over Z and paying attention to thefactor 4 above instead of passing to Q, one could obtain a stronger bound forg in some cases, but the authors have no evidence one way or the other.

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