topological invariants for equivariant flows: conley index

Institute of MathematicsPolish Academy of Sciences

Marcin Styborski

Topological invariants for equivariant flows:Conley index and degree

PhD Thesis

Supervisor: Dr hab. Marek IzydorekGdańsk University of Technology

Gdańsk 2009

Contents

Contents i

0 Introduction 10.1 Overview of the results . . . . . . . . . . . . . . . . . . . . . 2

1 Classical Conley’s theory 51.1 Morse–Conley–Zehnder equation . . . . . . . . . . . . . . . . 51.2 Continuation to a gradient . . . . . . . . . . . . . . . . . . . 91.3 Euler characteristic of the index . . . . . . . . . . . . . . . . 101.4 Conley index and the Brouwer degree . . . . . . . . . . . . . 10

2 LS -index 132.1 LS -flows and the index . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Cohomological LS -Conley index . . . . . . . . . . . 162.2 LS -index and the Leray–Schauder degree . . . . . . . . . . 18

2.2.1 Alternative approach in the particular case . . . . . . 21

3 Equivariant theory 253.1 Basic equivariant topology . . . . . . . . . . . . . . . . . . . 26

3.1.1 Orbit types . . . . . . . . . . . . . . . . . . . . . . . 273.1.2 Slices . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.3 G-complexes . . . . . . . . . . . . . . . . . . . . . . . 293.1.4 Euler ring U(G) . . . . . . . . . . . . . . . . . . . . . 30

3.2 Degree for G-equivariant gradient maps . . . . . . . . . . . . 323.3 Equivariant Conley index . . . . . . . . . . . . . . . . . . . . 343.4 Equivariant Morse–Conley–Zehnder equation . . . . . . . . . 37

3.4.1 Poincaré polynomial for an isolated orbit . . . . . . . 413.4.2 Some multiplicity results . . . . . . . . . . . . . . . . 45

3.5 Continuation of equivariant maps to a gradient . . . . . . . . 51

4 On the invertibility in U(G) 544.1 Technicalities . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Self-invertibility of SV . . . . . . . . . . . . . . . . . . . . . 55

5 Appendix 58

Bibliografy 62

Chapter 0

Introduction

About forty years have passed since Charles Conley defined the homo-topy index. Thereby, he generalized the ideas that go back to the calculusof variations work of Marston Morse. Within this long time the Conleyindex has proved to be a valuable tool in nonlinear analysis and dynamicalsystems. A significant development of applied methods has been observed.Later, the index theory has evolved to cover such areas as discrete dynam-ical systems, or analysis of flows defined on locally noncompact spaces cf.LS -index. Using the Conley theory, one is interested in the behavior ofthe particular sets of solutions, called isolated invariant sets, of differentialequations. The index of an isolated invariant set S is a homotopy type (or,in case of an LS -index, a stable homotopy type) of the quotient X/A ofa certain pair, called the index pair. It will be denoted by h(S). Since thehomotopy types cannot be lined up (like, for instance, the real numbers)and they are often very difficult to distinguish, they are fairly hard to workwith. Thus, the cohomological index H∗(X/A) has been found to be moreaccessible to the applications. It it easier to compare this index with otheralgebraic topological characteristics of the dynamical systems.

Probably the most important feature, among others, of the Conley indexis the invariance with respect to small perturbations of the initial differen-tial equation. A large collection of tools, called the homotopy invariants,has this special property. They include: the index of a zero of a vector field,topological degree, intersection number, Lefschetz number etc. Herein, weare focused on the topological degree and some of its extensions. The over-all aim of this thesis is to study the relationship between the degree ofa vector field and the Conley index of the induced flow. A large part ofthe thesis is devoted to the equivariant version of the Morse type inequali-ties (called equivariant Morse–Conley–Zehnder equation). The equivariantMorse inequalities have been used to compare the G-Conley index with thegradient equivariant degree. It was actually my primary intention. How-ever, this Morse–Conley–Zehnder equation seems to be very useful in thecritical point theory. Therefore, I decided to place some simple multiplicityresults for critical orbits of invariant (with respect to the Lie group action)functions. They are rather well known to many mathematicians. My in-tention was only to indicate possible directions in which one can go usingthese methods. Still, it would be interesting to extend these methods toinfinite-dimensional domains, i.e., to apply the G-equivariant MCZ equa-tion to critical point theory of strongly indefinite functionals on Hilbert

0.1 Overview of the results 2

spaces. From this point of view, it would allow us to investigate the Hamil-tonian systems using the variational treatment. As it is shown in [24], thisproblem is naturally equipped with symmetries of a group SO(2) (and itssubgroups).

0.1 Overview of the results.

The celebrated Poincaré-Hopf Theorem establishes a relationship be-tween the local invariants of a vector field at its zeroes and the global in-variants of the compact manifold where it is defined. The theses containedin this dissertation are another results of this kind. Here the compact mani-fold is replaced by a so-called index pair, and the topological degree plays arole of the local invariant of a vector field. We treat separately three cases.1. Firstly, we provide a known comparison of the classical Conley indexand the Brouwer degree (Theorem (1.17)). Namely, if φv is a (local) flowof the differential equation x = −v(x), and S is an isolated φv-invariant setwith an isolateing neighborhood N (cf. Section 1.1), then

χ(h(S)) = deg(v,N).

The above formula has been first proved by McCord in [33]. Earlier Dancerin [9] proved this kind of relation for considerably smaller class of isolatedinvariant sets, precisely for degenerate critical points. A simple proof canbe found in the book by Rybakowski [41] (See Chapter 3, Theorem 3.8).We present an elegant proof of this fact given by Razvan and Fotouhi in[37], based on Morse inequalities and Reineck continuation theorem [38].2. The LS -Conley index, the extension of the Conley’s invariant, is pre-sented, and the relations to the Leray–Schauder degree are studied. Theextension of the classical Conley’s theory was introduced by Gęba, Izydorekand Pruszko in [16]. They considered so-called LS -vector fields in a Hilbertspace, i.e., completely continuous perturbations of a bounded linear opera-tor L : H → H, and defined the index for flows induced by such maps. Thereis a particular property that makes the LS -index applicable to many varia-tional problems. Namely, an operator L can be strongly indefinite, i.e., bothpositive and negative eigenspaces of L can be infinite-dimensional. Furtherdevelopment of this homotopy invariant was presented by Izydorek in [23].He defined a cohomological LS -index and using this index gave existenceresults for various strongly indefinite problems. We briefly sketch out thisdefinition. A cohomological version of the LS -index allows us to define theBetti numbers and the Euler characteristic of the LS -index in the mostnatural way.

Let H be a real, infinite-dimensional Hilbert space. With a locally Lips-chitz vector field f : H → H, which is a completely continuous perturbation,say K, of the bounded (invertible) linear operator L, one can associate a


local flow φtf satisfying

d

dtφtf = −f φtf , φ0

f = id.

Under certain assumptions we prove the formula (cf. Theorem (2.13))

χ(hLS (S)) = degLS (f , N).

The right-hand side of the above equality stands for the standard Leray–Schauder degree with respect to a bounded set N . The map f is defined byf(x) = x+L−1K(x). On the other hand we have the Euler characteristic (cf.Definition (2.11)); hLS (S) stands for the LS -index of an isolated invariantset S = inv(N) of the flow φtf . The proof is based on finite-dimensionalformula mentioned above. Similar result was obtained by Kryszewski andSzulkin in [31] for S being a critical point of a smooth strongly indefinitefunctional.

We also give an alternative method of proving Theorem (2.13), at leastin a particular case, see Theorem (2.22). The isolated invariant set is theorigin of a Hilbert space, and a map L is of the form Lx := x+− x−, wherex = x+ + x− ∈ H = H+ ⊕H−.3. At last, the G-equivariant Conley index and the G-equivariant gradientdegree is studied. We placed the most emphasis on this case. With thisend in view we proved an equivariant version of Morse–Conley–Zehnderequation, see Theorem (3.47). The key point to obtain this result wasto accurately define the Poincaré polynomial of the G-index hG(S) of anisolated invariant G-set. Having in mind the form of the elements of theEuler ring (cf. Proposition (3.16)), we define

PG(t, hG(S)) :=∑

(H)∈Φ(G)

∞∑q=0

rankHq(X(H)/G, (X>(H) ∪ A(H))/G) tq uG(H),

where (X,A) is an arbitrary G-index pair for S. Beside the fact that thispolynomial has an unfriendly form, and causes some technical difficulties,the proof of the Morse–Conley–Zehnder equation is a consequence of stan-dard cohomological arguments. With the help of the prepared tools, andusing approximation techniques for gradient G-equivariant mappings pro-vided by Gęba (see Theorem (3.24)), we prove that (vide Theorem (3.61))

u(hG(S)) = deg∇G(f,Ω),

where u stands for, roughly speaking, an equivariant Euler characteristic(taking values in the Euler ring U(G)).

In addition to the comparison of the Conley index and the degree, onecan also find two, unconnected at the first sight, results. The first one isconcerned with the G-flows and asserts that an isolated invariant set of a


G-flow can be linked by a special kind of homotopy to the isolated invariantset of a gradient G-flow (cf. Theorem (3.74)). This is the G-equivariantcounterpart of the Reineck continuation theorem [38]. Using it I proved thatthe G-equivariant Conley index is a homotopy type of a finite G-complex (cf.Corollary (3.79)). The second one is rather purely algebraic. Gołębiewskaand Rybicki showed in [18] that if V is a finite dimensional orthogonal G-representation, then u(SV ) is an invertible element in the ring U(G) (seeDefinition (3.15)). By strengthening the assumptions I give a very simple,geometric proof of this result. Namely, I assume that G is finite and abelian.At the same time I show that u(SV ) is self-invertible, i.e., u(SV )−1 = u(SV ).

As already mentioned, some multiplicity results are also given (cf. Propo-sitions (3.66), (3.68) and (3.73)). These results provide an estimation frombelow of the number of critical orbits of G-invariant functions, for G = Zp(p being a prime number), and G = SO(2). The case of the most generalZ2-action on an inner product space of finite dimension is presented. Themain argument is the equivariant Morse–Conley–Zehnder equation.

Acknowledgements.

First and foremost I would like to express my gratitude to my advisor,prof. Marek Izydorek for his professional guidance and constant encourage-ment which made this work possible. I am very grateful for his suggestionsand many hours of enlightening discussions.

I would like to thank prof. Kazimierz Gęba for always having his dooropen to answer my questions. I also express my gratitude to all the partic-ipants of the seminar on Topological Methods in Nonlinear Analysis whichtakes place at Gdańsk University of Technology for friendly criticism of mypresentations.

The work was supported in part by MNiSW grant N N201 273235.

Chapter 1

Classical Conley’s theory

The purpose of this chapter is to provide the rudiments of Conley’s indextheory for flows on locally compact metric spaces. The reader will find herebasic definitions which will be needed in the further parts of the thesis. Thematerial is presented in such a way that we will be able to give a completeproof of the formula joining the Conley index and the topological degree.

1.1 Morse–Conley–Zehnder equation.

Let X be a locally compact metric space. Recall, that a continuous mapφ : D → X is called a local flow onX if the following properties are satisfied:

• D is an open neighborhood of 0 ×X in R×X;

• for each x ∈ X there exist αx, ωx ∈ R ∪ ±∞ such that (αx, ωx) =t ∈ R; (t, x) ∈ D;

• φ(0, x) = x and φ(s, φ(t, x)) = φ(s + t, x) for all x ∈ X and s, t ∈(αx, ωx) such that s+ t ∈ (αx, ωx).

In the case of D = R×X, we call φ the flow on X. We will interchangablyuse φt(x) and φ(t, x). Thus, we have φ0 = idX , and φt+s = φt φs. Themain objects of this theory are isolated invariant sets and associated withthem isolating neighborhoods. Let φt be a flow on X. A subset S of Xis called an invariant set, if S =

⋃t∈R φ

t(S). For N ⊂ X we define themaximal invariant set contained in N :

inv(N) :=x ∈ N ; φt(x) ∈ N, t ∈ (αx, ωx)

.

If N is compact and inv(N) ⊂ int(N), then N is called an isolating neigh-borhood, and S = inv(N) is an isolated invariant set.

Let N be a compact subset of X. We say that L ⊂ N is positivelyinvariant relative to N if for any x ∈ L the inclusion φ[0,t](x) ⊂ N impliesthat φ[0,t](x) ⊂ L.

(1.1) Definition (Index pair). A compact pair (N,L) is called an indexpair for S, if:

• N \ L is a neighborhood of S and S = inv(N \ L

);

• L positively invariant relative to N ;

1.1 Morse–Conley–Zehnder equation 6

• if x ∈ N and there exists t > 0, such that φt(x) 6∈ N , then there existss ∈ [0, t], such that φs(x) ∈ L.

The next two theorems are crucial in the definition of homotopy Conleyindex. The proofs can be found in Salamon’s paper [45]

(1.2) Theorem. Every isolated invariant set S admits an index pair (N,L).

If (N,L) is a pair of spaces, L ⊂ N , then the quotient N/L is obtainedfrom N by collapsing L to a single point denoted by [L], the base point ofN/L. A set X ⊂ N/L is open if either X is open in N and X ∩ L = ∅ orthe set (X ∩N \ L) ∪ L is open in N .

Recall that f : (X, x0)→ (Y, y0) is a homotopy equivalence if there existsa map g : (Y, y0) → (X, x0) such that g f is homotopic to id|X rel. x0

and f g is homotopic to id|Y rel. y0. If there is a homotopy equivalencef : (X, x0)→ (Y, y0) we say that the pairs (X, x0) and (Y, y0) are homotopyequivalent or they have the same homotopy type. The homotopy type of(X, x0) is denoted by [X, x0].

(1.3) Theorem. Let (N0, L0) and (N1, L1) be two index pairs for the iso-lated invariant set S. Then the pointed topological spaces N0/L0 and N1/L1

are homotopy equivalent.

(1.4) Definition. If (N,L) is any index pair for the isolated invariant setS, then the homotopy type h(S, φt) = [N/L] is said to be the Conley (ho-motopy) index of S. When the flow is clear from context, we just writeh(S) for short.

Theorem (1.3) says that h(S) is independent of the choice of an indexpair. Let us illustrate the concept of Conley index by the following simpleexample.

(1.5) Example. Let Ω ⊂ Rn be an open and bounded set and f : Rn → Rbe a smooth function such that (∇f)−1(0) ∩ ∂Ω = ∅. The smoothness off implies that ∇f is a locally Lipschtz continuous map, and hence by thetheorem of Picard-Lindelöf the equation d

dtu(t) = ∇f(u(t)) defines a local

flow on Ω: φtf (x) = u(t), where u : (αx, ωx) → Ω is a solution curve of theabove equation passing through x at t = 0, and defined on its maximalinterval of existence (αx, ωx). The rest points of φtf are the critical pointsof f . They are hyperbolic if f is a Morse function, i.e., the Hessian of fis nonsingular at every x ∈ Crit(f), where Crit(f) = x ∈ Rn; ∇f(x) = 0.In this case the number

indf (x) = #negative eigenvalues of the Hessian∇2f(x)

is well defined. The Conley index of an isolated invariant set S = x,where x ∈ Crit(f), is the homotopy type of a pointed k-sphere, wherek = n− indf (x).

This example shows that the Conley index and the Morse index giveus the same qualitative information about the flow near the critical point,


whenever the latter is defined.Below, we state the basic properties of the Conley index. Let 0 denote

the homotopy type of a pointed one point space.

(1.6) Proposition (cf. [5]). Let φt : X → X be a local flow on a locallycompact metric space and let N be an isolating neighborhood for φt withS = invN . The Conley index h(S) has the following properties:

Nontriviality If h(S) 6= 0, then S 6= ∅;

Summation formula If S = S1∪S2 is a disjoint union of isolated invari-ant sets, then h(S) = h(S1)∨h(S2) (wedge sum; for the definition seepage 15);

Multiplication formula If Si is an isolated invariant set of a local flowφti : R×Xi → Xi, i = 1, 2 then S = S1×S2 is a isolated invariant setof φt = φt1 × φt2 : R×X1 ×X2 → X1 ×X2 and h(S, φt) = h(S1, φ

t1) ∧

h(S2, φt2) (smash product; for the definition see page 15).

In what follows we restrict ourselves to consider flows instead of localflows. If φ is a local flow on X defined by a vector field (integral curvesof differential equation), then one can replace the initial vector field by thecompactly supported one, such that the initial and the new map coincideon isolating neighborhood.

Let φt be a flow on X. For x ∈ X define its α-limit and ω-limit sets asfollows:

α(x) :=⋂t≥0

φ(−∞,−t](x), ω(x) :=⋂t≥0

φ[t,+∞)(x).

(1.7) Definition. A Morse decomposition of an isolated invariant set S isa finite collection M (S) = Mi; 1 ≤ i ≤ l of subsets Mi ⊂ S, which aredisjoint, compact and invariant, and which can be ordered (M1,M2, . . . ,Ml)so that for every x ∈ S \ ⋃1≤j≤lMj there are indices i < j such that

ω(x) ⊂Mi, α(x) ⊂Mj.

Notice that in the previous example, the set Crit(f) of all critical pointsof f forms a Morse decomposition of inv(Ω). Indeed, we can arrange thecritical points x1, . . . , xm in the following manner: i < j whenever f(xi) >f(xj).

Assuming that all groups Hq(A,B) have a finite rank for all q ≥ 0, definethe formal power series

P(t, A,B) =∞∑q=0

rankHq(A,B) · tq

called the Poincaré series of pair (A,B). If the pair (X,A) is of finite type,i.e., Hq(A,B) = 0 for q ≥ q0, then we say that P(t, A,B) is a Poincaré


polynomial of (A,B). One can prove that for an isolated invariant set thereis an index pair (N,L) for which the isomorphism H∗(N,L) ∼= H∗(N/L)holds. Such an index pair is called regular (cf. [36, 45]). We can thereforedefine the Poincaré polynomial1 of h(S) as

P(t, h(S, φt)) := P(t, N, L)

where (N,L) is any regular index pair for S.The following theorem generalizes the classical Morse inequalities that

give the lower bounds for the number of critical points of a smooth functionon a compact oriented closed manifold M (cf. [6, 42]).

(1.8) Theorem (Morse–Conley–Zehnder equation, cf. [6, 42]). If S is anisolated invariant set with a Morse decomposition M (S) = Mi; 1 ≤ i ≤ l,then there is a polynomial Q with nonnegative coefficients such that

(1.9)l∑

i=1

P(t, h(Mi)) = P(t, h(S)) + (1 + t)Q(t).

(1.10) Example (cf. Definition 1.1, Example 1.8 and Theorem 1.15 in[7]). The Lyusternik–Schnirelmann category of a space X is the smallestcardinality of an open covering of X by contractible subsets. For the torusT 2 = R2/Z2, it equals 3. It is the lower bound of the number of criticalpoints that a smooth real–valued function on a torus could possess. Onecan check that the function F (x, y) = sin πx sin πy sin π(x + y) has exactlythree critical points. The standard example with a hight function on a torusshows that there is a function with four nondegenerate critical points (cf.[34]). The question is: Does a Morse function f : T 2 → R with preciselythree critical points exist? The answer is negative and comes immediatelyfrom Theorem (1.8).

To see this, assume that f : T 2 → R is a Morse function with threecritical points. The possible indices that occur in this situation are: (i)2, 2, 0 or (ii) 2, 0, 0 or (iii) 2, 1, 0. Let us consider only the third case. Theargument for both (i) and (ii) is exactly the same. Consider the negativegradient flow of f . The critical points of f form a Morse decompositionof the torus and, as we have seen in example (1.5) the Conley indices arehomotopy types of pointed spheres of dimension 2, 1, 0, respectively. Hencethe left hand side of (1.9) is t2 + t + 1. Obviously, the pair (T 2, ∅) is anindex pair for T 2, and P(t, h(T 2)) = t2 + 2t+ 1. Applying (1.9), we obtainthe equality

t2 + t+ 1 = t2 + 2t+ 1 + (1 + t)Q(t),

which cannot be true.1We will see later (cf. Proposition (1.15) and Corollary (1.16)) that h(S) is a homotopy

type of a finite CW -complex. Therefore, in view of isomorphism H∗(N,L) ∼= H∗(N/L),the pair (N,L) is of finite type.

1.2 Continuation to a gradient 9

1.2 Continuation to a gradient.

Let φ : R × X × [0, 1] → X be a continuous family of flows on X, i.e.,φtλ := φ(t, · , λ) : X → X is a flow on X. Suppose that N ⊂ X is compactand Si = inv(N, φti), i = 0, 1. We say that two isolated invariant sets S0 andS1 are related by continuation, or S0 continues to S1, if for all φtλ λ ∈ [0, 1],N is an isolating neighborhood. The notion of continuation is essential inthe Conley index theory, as demonstrated by the following theorem.

(1.11) Theorem ([5]). If S0 and S1 are related by continuation, then theirConley indices coincide.

Recall that a Morse–Smale gradient flow satisfies the following (i) allbounded orbits are either critical points of the potential function or orbitsconnecting two critical points; (ii) stable and unstable manifolds of the restpoints intersect transversally.

Let Ω ⊂ Rn be an open set, F : Ω → Rn a smooth vector field and letφtF : Ω→ Ω be a flow generated by x(t) = −F (x(t)). Assume that N is anisolating neighborhood and S = inv(N).

(1.12) Theorem (Reineck [38]). Set S can be continued to an isolatedinvariant set of a positive gradient flow of certain function f defined on theopen set U containing N and without changing F on Ω \N . Moreover, thiscan be done in such a way that the new flow is Morse–Smale.

Remarks. The fact that such function f exists has been proved by Robbinand Salamon in [39]. They showed that for an isolated invariant set S =invN there exists a smooth function f : U → R defined on a neighborhoodof N such that2

• f(x) = 0 iff x ∈ S and

• ddt|t=0f(φt(x)) < 0 for all x ∈ Ω \ S.

The function which satisfies this conditions is called the Lyaponov function.In general, one cannot expect that for an isolated invariant set its Lyaponovfunction would have only nondegenerate critical points, i.e., the rest pointsof gradient flow are hyperbolic. By the Kupka–Smale theorem (cf. Theorem6.6 in [2]) this can be obtained via arbitrary small perturbation of ∇f inthe C1-topology. Hence, without loss of generality, we can assume that thegradient flow is Morse–Smale.

Following Reineck, one can explicitly write the homotopy connecting −Fand ∇f . Define h : Ω× [0, 1]→ Rn by setting

(1.13) h(x, λ) = ρ(x)[λ∇f(x) + (λ− 1)F (x)] + (ρ(x)− 1)F (x),

where ρ : Ω → [0, 1] is a smooth function which equals 1 on a compactneighborhood of S (denoted M , M ⊂ int(N)); and ρ is zero on Ω \N .

2Cf. Lemma (3.75) on page 52

1.3 Euler characteristic of the index 10

We have adopted the Reineck theorem to the equivariant case. The proofis presented in Section 3.5.

1.3 Euler characteristic of the index.

Recall that the Euler characteristic of a topological pair (X,A) is definedas

(1.14) χ(X,A) =∞∑q=0

(−1)q rankHq(X,A),

provided that pair (X,A) is of a finite type. Notice that χ(X,A) is equalto P(−1, X,A). If both Hq(X) and Hq(A) are finitely generated (e.g.if X and A are CW-complexes), the integer χ(X,A) is well defined. Inparticular, if A is a point in X (that is, X is a pointed space), then we haveχ(X, pt) =

∑∞q=1(−1)q rank Hq(X), where Hq(X) stands for the reduced

cohomology. In particular, the Euler characteristic is well defined for theConley index of an isolated invariant set.

The next proposition, due to Gęba (cf. [14], Proposition 5.6.), says thatone can always choose a nice space from the homotopy class, namely a finiteCW-complex.

(1.15) Proposition. Let N be an isolating neighborhood for a gradientMorse–Smale flow φt on Rn. Then h(inv(N), φt) is a homotopy type offinite CW-complex.

(1.16) Corollary. Let N be an isolating neighborhood for a flow φt on Rn

generated by x = −F (x). Then h(inv(N), φt) is a homotopy type of finiteCW-complex.

Proof. Since inv(N) is related by continuation to some isolated invariant setof gradient Morse–Smale flow, the result follows from Proposition (1.15).

1.4 Conley index and the Brouwer degree.

The connection between the Conley index and the topological degreeare noticeable at first glance. The homotopy invariance of the Brouwerdegree corresponds to the continuation property of the Conley index. Theexistence axiom refers to nontriviality property which says that nontrivialindex implies nonempty isolated invariant set. The next common feature ofboth invariants is that they are determined by a behaviour of a vector field(flow) on a boundary of the set under investigation.

The first quite general result concerned with the relationship betweenhomotopy invariant (local indices of zeros) of vector field and the Conleyindex has been proved by McCord in [33]. Earlier Dancer in [9] proved theformula χ(h(x , φtf )) = deg(f, Ux), where x is a degenerate rest point ofa gradient flow of −f and Ux stands for its neighborhood. He thoroughly

1.4 Conley index and the Brouwer degree 11

discussed the Betti numbers of index h(x , φtf ). We shall present theproof of a more general fact, where the degenerate critical point in Dancer’sformula is replaced by an isolated invariant set and the flow is not necessarilygradient.

We will briefly recall the notion of the degree of a map. Let Ω ⊂ Rn bean open and bounded set. If f : Ω→ Rn is a continuous map and does notvanish on the boundary ∂Ω, then it is well known that there is an integerdeg(f,Ω) ∈ Z, called the Brouwer degree (cf. for instance [32, 44]). Itsatisfies the following axioms:

Nontriviality If 0 ∈ Ω then deg(I,Ω) = 1, where I is an identity map;

Existence If deg(f,Ω) 6= 0 then f−1(0) ∩ Ω is nonempty;

Additivity If Ω1,Ω2 are open, disjoint subsets of Ω and there is no zerosof f in the complement Ω \ (Ω1 ∪ Ω2), then

deg(f,Ω) = deg(f,Ω1) + deg(f,Ω2);

Homotopy invariance If h : Ω × [0, 1] → Rn is a continuous map suchthat h(x, t) 6= 0 for all (x, t) ∈ ∂Ω× [0, 1], then

deg(h( · , 0),Ω) = deg(h( · , 1),Ω)

There is a generic situation, when the degree is easy to calculate. If ϕ : Ω→R is a Morse function such that deg(∇ϕ,Ω) is defined, then

deg(∇ϕ,Ω) =∑

x∈(∇ϕ)−1(0)∩Ω

(−1)indϕ(x).

(1.17) Theorem (cf. [37]). Let F : Ω→ Rn be a locally Lipschitz map anddenote by φtF the local flow generated by x = −F (x). If N is an φtF -isolatingneighborhood and S = inv(N) then

(1.18) χ(h(S)) = deg(F, int(N)).

In what follows we will use deg(F,N) instead of deg(F, int(N)).

Proof. By the Reineck continuation theorem, S continues to an isolatedinvariant set of a Morse–Smale gradient flow φtf , that consists of only non-degenerate critical points of f and of connecting orbits between them. De-note this set by S ′. By the continuation property of the Conley indexh(S) = h(S ′). The set of critical points x1, . . . , xm forms a Morse de-composition of S ′, and by Example (1.5) one has that h(xi, φtf ) is thehomotopy type of a pointed k-sphere, where k = n − indf (xi). Hence, thePoincaré polynomial of h(xi, φtf ) is of the form

(1.19) P(t, h(xi, φtf )) = tn−indf (xi).

1.4 Conley index and the Brouwer degree 12

Applying Theorem (1.8) one obtains

χ(h(S)) = χ(h(S ′)) = P(−1, h(S ′))

=m∑i=1

P(−1, h(xi, φtf )) = (−1)nm∑i=1

(−1)indf (xi)(1.20)

For 1 ≤ i ≤ m, let Ωi be a neighborhood of xi in N such that Ωi ∩ Ωj = ∅.Using the homotopy invariance of the Brouwer degree and the additivityproperty leads to

(1.21) deg(−F,N) = deg(∇f,N) =m∑i=1

deg(∇f,Ωi).

Now it is easy to compute deg(∇f,Ωi). Since f is a Morse function, thehessian ∇2f(xi) is a non-degenerate linear operator. The degree of ∇fwith respect to Ωi is (−1)µ, where µ is the number of negative eigenvaluesof ∇2f(xi). That is, deg(∇f,Ωi) = (−1)indf (xi). By (1.21) we obtain

(1.22) deg(F,N) = (−1)n deg(−F,N) = (−1)nm∑i=1

(−1)indf (xi)

Combining (1.20) and (1.22) we obtain formula (1.18).

(1.23) Example. The simplest example for Theorem (1.17) is given by theequation x = x on Rn. Hence the vector field is −id : Rn → Rn, and itsdegree with respect to the unit ball depends on the dimension n, and equals(−1)n. The origin is an isolated equilibrium with an index pair (Dn, Sn−1).The Euler characteristic of an index is obviously χ(Dn/Sn−1) = (−1)n.

(1.24) Example. The map F : R2 → R2

F (x, y) := (−x− y + x(x2 + y2), x− y + y(x2 + y2))

gives us a little bit more refined illustration. The annulus

A =

(x, y) ∈ R2; r ≤ x2 + y2 ≤ R

0 < r < 1 < R

is an isolating neighborhood. Indeed, the inner product 〈F (x, y), (x, y)〉 =(x2 +y2)2−(x2 +y2) shows that for x2 +y2 < 1 the vector field points insidethe annulus, while for x2 + y2 > 1 the vectors point outside of it. The exitset is a disjoint union of the boundary circles. The index is a homotopy typeof a wedge sum S2 ∨ S1. It is easily seen that S2 ∨ S1 is composed of 0-,1-,and 2-dimensional cells. Hence the Euler characteristic modulo a basepointequals zero. The additivity property of the Brouwer degree implies quicklythat deg(F,A) = 0.

Chapter 2

LS -index

The homotopy invariant called the LS -index is a generalization of theclassical homotopy index introduced by Conley. The construction of theLS -index presended in [16], based on the Galerkin-type approximation,reminds the way the Leray–Schauder degree extends the classical Brouwerdegree. This extension originated from the application of the index to theHamiltonian dynamics. Searching for periodic solutions of Hamiltonian sys-tems is converted into a problem of finding critical points of certain actionfunctional Φ: H → R defined on infinite dimensional real Hilbert space.Moreover, Φ turns out to be strongly indefinite, i.e., the gradient flow ofΦ has an infinite dimensional both stable and unstable manifolds, so theclassical Morse theory approach cannot be used. It is worthwhile to men-tion that the LS -index has been also successfully applied by Izydorek andRybakowski in the study of strongly indefinite elliptic systems cf. [26, 28].

The purpose of this chapter is to provide the basic facts about theLS -index and to prove the formula relating the index to the Leray–Schauderdegree. In Subsection 2.1.1 we give the definition of the Betti numbers andEuler characteristic of an LS -index. Theorems (2.13) and (2.15) are crucialin this chapter.

2.1 LS -flows and the index.

Let H be a real, separable Hilbert space, and L : H → H be a linearbounded operator which satisfies the following assumptions:

(L.1) L gives a splitting H =⊕∞n=0Hn onto finite dimensional, mutually

orthogonal L-invariant subspaces;

(L.2) L(Hn) = Hn for n > 0 and L(H0) ⊂ H0, where H0 is a subspacecorresponding to the part of spectrum on imaginary axis, i.e., σ0(L) :=σ(L|H0) = σ(L) ∩ iR;

(L.3) σ0(L) is isolated in σ(L).

It is possible that dimH± = ∞, where H− (resp. H+) is an invariantsubspace corresponding to those parts of spectrum of L which lie on the left(resp. right) half complex plane. Operators with the above property arecalled strongly indefinite.

2.1 LS -flows and the index 14

Let Λ be a compact metric space. A family of flows indexed by Λ is acontinuous map φ : R×H × Λ→ H such that φλ : R×H → H defined byφλ(t, x) = φ(t, x, λ) is a flow on H. As before we write φt(x, λ) instead ofφ(t, x, λ). If X ⊂ H and φ is a family of flows indexed by Λ then we define

inv(X × Λ) = inv(X × Λ, φ) :=

(x, λ) ∈ X × Λ; φt(x, λ) ∈ X, t ∈ R.

(2.1) Definition. A family of flows φt : H × Λ → H is called a family ofLS -flows if

φt(x, λ) = etLx+ U(t, x, λ),

where U : R×H × Λ→ H is completely continuous.

Recall, that a map is completely continuous if it is continuous and mapsbounded sets to relatively compact sets.

(2.2) Definition. We say that a map f : H × Λ → H is a family of LS -vector fields, if f is of the form

f(x) = Lx+K(x, λ), (x, λ) ∈ H × Λ,

where K : H ×Λ→ H is completely continuous and locally Lipschitz map.

If in the above definitions Λ = λ0, we drop the parameter space outfrom notation, and we say that f is an LS -flow or an LS -vector field.

Suppose that f : H → H is an LS -vector field, f(x) = Lx+K(x). Wesay that f is subquadratic if |〈K(x), x〉| ≤ a ‖x‖2 + b for some a, b > 0. Onecan prove that if f is subquadratic then f generates an LS -flow (cf. [23]).That is for all x ∈ H, there exists a C1-curve

φ(·)(x) : R→ H

satisfyingd

dtφt(x) = −f φt(x), φ0(x) = x,

and is of the form φt(x) = e−tLx + U(t, x), where U : R × H → H iscompletely continuous. Without loss of generality we will restrict our con-sideration to subquadratic LS -vector fields (cf. [16, 23]).

An isolating neighborhood for a flow φt on infinite dimensional space isdefined similarly to finite dimensional case. The difference lies in the factthat we cannot expect compactness of that set.

(2.3) Definition. A bounded and closed set N is an isolating neighborhoodfor a flow φt if and only if inv(N) ⊂ int(N).

The isolating neighborhoods are stable with respect to small perturba-tion of the flow. The sense of this concept is given by the following.

(2.4) Proposition (Gęba et al. [16]). Let φ : R×H × Λ→ H be a familyof LS -flows. For any bounded and closed N ⊂ H the set

Λ(N) = λ ∈ Λ; inv(N, φλ) ⊂ int(N)

is open in Λ.


The key feature of the LS -flows is the following compactness property.

(2.5) Proposition (Gęba et al. [16]). Let Λ be a compact metric spaceand let φ : R×H × Λ→ H be a family of LS -flows. If N is a closed andbounded, then S := inv(N × Λ) is a compact subset of N × Λ.

We are going to work in the category of compact metrizable spaces witha base point. The notion f : (X, x0)→ (Y, y0) means that f is a continuousmap preserving base points, i.e., f(x0) = y0. The Cartesian product isdefined in this category by (X, x0)× (Y, y0) = (X × Y, (x0, y0)). The wedgeof two pointed spaces, i.e., the space X ∨Y = X×y0∪x0×Y is closedin X × Y . Hence, the smash product X ∧ Y = (X × Y )/(X ∨ Y ) is alsoan object in that category. In addition, if f : X → Y and g : X ′ → Y ′ thenf ∧ g : X ∧X ′ → Y ∧ Y ′ is defined.

Consider the circle as the unit interval modulo its end points S1 =[0, 1]/0, 1. The suspension functor is defined to be the smash productSX := S1 ∧X. For any m ∈ N we define SmX := S(Sm−1X).

Let ν : N∪0 → N∪0 be a fixed map and suppose that (En)∞n=n(E) isa sequence of spaces and (εn : Sν(n)En → En+1)∞n=n(E) is a sequence of maps.

(2.6) Definition. We say that a pair E = ((En)∞n=n(E), (εn)∞n=n(E)) is aspectrum if there exists n0 ≥ n(E) such that εn : Sν(n)En → En+1 is ahomotopy equivalence for all n ≥ n0.

One can define the notion of maps of spectra, homotopy of spectra,their homotopy type etc. For us it is sufficient to know that a homotopytype [E] of a spectrum E is uniquely determined by a homotopy type of apointed space En for n sufficiently large. Moreover, in order to define thehomotopy type [E] one only needs a sequence (En)∞n=n(E) such that Sν(n)Enis homotopy equivalent to En+1 for n sufficiently large.

Assume that f : H → H is an LS -vector field, f(x) = Lx+K(x). Letφt : H → H be the LS -flow generated by f and assume that N ⊂ H isan isolating neighborhood for φt. Denote by Pn : H → H the orthogonalprojection onto Hn =

⊕ni=1 Hi. Set H−n := H− ∩ Hn and H+

n := H+ ∩ Hn

and definefn : Hn → Hn, fn(x) = Lx+ PnK(x).

Let φtn : H → H be a flow induced by fn. The definition of LS -Conleyindex is based on the following.

(2.7) Lemma (Gęba et al. [16]). There exists n0 ∈ N such that Nn =N ∩Hn is an isolating neighborhood for a flow φtn provided that n ≥ n0.

By the above lemma the set Sn := inv(Nn, φtn) is an isolated and in-variant (by definition) and thus admits an index pair (Yn, Zn) by Theorem1.2. The Conley index of Sn is the homotopy type [Yn/Zn]. Fix a mapν : N ∪ 0 → N ∪ 0 by setting ν(n) := dimH−n+1. Using the continu-ation property of the Conley index one can prove that the pointed spaceYn+1/Zn+1 is in fact homotopy equivalent to the ν(n)-fold suspension of


Yn/Zn, that is[Yn+1/Zn+1] = [Sν(n)(Yn/Zn)]

for all n ≥ n0. The sequence (En)∞n=n0= (Yn/Zn)∞n=n0

represents the spec-trum, say E and uniquely determines its homotopy type [E]. This leads usto the definition.

(2.8) Definition. Let φt be an LS -flow generated by an LS -vector field.If N is an isolating neighborhood for φt and S := inv(N, φt), then thehomotopy type of spectrum

hLS (S, φt) := [E]

is well defined and we call it the LS -Conley index of S with respect to φt.When the flow is clear from context we just write hLS (S).

Let 0 represents the homotopy type of spectrum such that for all n ≥ 0En consists of a distinguished point and εn maps the point of En into thepoint in En+1.

(2.9) Proposition (Gęba et al. [16]). The LS -Conley index has the fol-lowing properties:Nontriviality Let φt : H → H be an LS -flow and N ⊂ H be an isolating

neighborhood for φt with S := inv(N). If hLS (S) 6= 0, then S 6= ∅;

Continuation Let Λ be a compact, connected and locally contractible met-ric space. Assume that φt : H × Λ → H is a family of LS -flows.Let N be an isolating neighborhood for a flow φtλ for some λ ∈ Λ andSλ := inv(N, φtλ). Then there is a compact neighborhood Uλ ⊂ Λ suchthat

hLS (Sµ) = hLS (Sν)

for all µ, ν ∈ Uλ.

2.1.1 Cohomological LS -Conley index. The main reference for thissection is [23]. Now and subsequently H∗ denotes the Alexander–Spaniercohomology functor. Let E = (En, εn)∞n=n(E) be a spectrum. Define ρ : N ∪0 → N ∪ 0 by setting ρ(0) = 0 and ρ(n) =

∑n−1i=0 ν(i) for n ≥ 1. For a

fixed q ∈ Z consider a sequence of cohomology groups

Hq+ρ(n)(En), n ≥ n(E).

Denote by S∗ : Hq(X) → Hq+1(SX) the suspension isomorphism. Definea sequence of homomorphisms hnn≥0 such that the following diagramcommutes

Hq+ρ(n+1)(En+1)hn - Hq+ρ(n)(En)

Hq+ρ(n+1)(Sν(n)En)(S∗ )−ν

(n)

-

ε q+ρ(n+

1)

n

-


Thus we see that Hq+ρ(n)(En), hn forms an inverse system and we areready to make the following definition.

(2.10) Definition. The qth cohomology group of a spectrum E is definedto be

CHq(E) := lim←−Hq+ρ(n)(En), hn.

Since En+1 is homotopically equivalent to Sν(n)En for n ≥ n0, we seethat

hn : Hq+ρ(n+1)(En+1)→ Hq+ρ(n)(En)

is an isomorphism for n ≥ n0 and the sequence of groups Hq+ρ(n)(En)stabilizes. This simple observation implies that:

• CHq(E) ∼= Hq+ρ(n)(En) for n ≥ n0;

• the graded group CH∗(E) is finitely generated if H∗(En0) is finitelygenerated;

• the spectrum E is of finite type if the space En0 is of finite type.

These groups may be nonzero for both positive and negative q’s (cf. [23]or Example (2.25)).

Now we are able to define the Betti numbers and the Euler characteristicof an LS -Conley index represented by the spectrum E in the obvious way.

(2.11) Definition. Let E be a fixed spectrum. The qth Betti number of Eis defined as

βq(E) := rankCHq(E),

and the Euler characteristic is given by

χ(E) :=∑q∈Z

(−1)qβq(E).

(2.12) Remark. There exist n0 such that for all n ≥ n0 we have χ(E) =(−1)ρ(n)χ(En).

Proof. Since CHq(E) ∼= Hq+ρ(n)(En) for n ≥ n0 we have

(−1)ρ(n)χ(E) = (−1)ρ(n)∑q∈Z

(−1)qβq(E)

=∑q∈Z

(−1)q+ρ(n)βq+ρ(n)(En) = χ(En).

2.2 LS -index and the Leray–Schauder degree 18

2.2 LS -index and the Leray–Schauder degree.

In this section we shall prove an infinite dimensional counterpart of The-orem (1.17). The main idea is to use a finite dimensional approximationand the proof is in fact based on the finite dimensional formula (1.18). Forbetter clarity we divide the result into two separate statements. Firstly, wewill present a proof for maps being completely continuous perturbations ofan isomorphism L : H → H, and next we show how to weaken the assump-tion about the linear part. Namely, we assume that L is merely a selfadjointoperator.

The theorem generalizes the result of Chang (Theorem 3.3. in [4]) aswell as the result of Kryszewski and Szulkin (Theorem 6.1. in [31]). Theyconsidered a gradient mappings on Hilbert spaces (manifolds) satisfying aso-called Palais–Smale condition, and compared the LS -degree with theEuler characteristic of a Gromoll–Mayer pair of the critical point. The lasttwo authors generalized Chang’s result for a strongly indefinite functionals.In the next theorem we do not restrict our attention to the gradients, norto critical points of functionals as isolated invariant sets.

Let U be an open and bounded subset of H. Denote by degLS (f, U) theLeray–Schauder degree, defined for completely continuous perturbations ofan identity, i.e., f(x) = x+F (x), where F is completely continuous such thatF (x) 6= x on ∂U . For more details about the degree theory we refer to thebook by Lloyd [32]. Consider an LS -vector field f in H, f(x) = Lx+K(x),where L is strongly indefinite linear bounded and invertible operator, andK is a completely continuous map. Suppose that f does not vanish on ∂U .We will define the degree for the class of such maps in the following manner:

degL(f, U) := degLS (I + L−1K,U).

Since the zero sets for both f and I + L−1K are the same, and L−1K iscompletely continuous, the above definition works. The degL inherits allthe properties of the Leray–Schauder degree. In particular one has:

Nontriviality If 0 ∈ U then degL(L,U) = 1;

Existence If degL(f, U) 6= 0 then f has a zero inside U ;

Additivity If U1, U2 are open, disjoint subsets of U and there are no zerosof f in the completion U \ (U1 ∪ U2), then

degL(f, U) = degL(f, U1) + degL(f, U2);

Homotopy invariance If h : H × [0, 1] → H is an LS -vector field forall t ∈ [0, 1] such that h(x, t) 6= 0 for all (x, t) ∈ ∂U × [0, 1], thendegL(h( · , t), U)) is independent of t ∈ [0, 1].

The following theorem is the main part of the author’s paper [47].


(2.13) Theorem. Assume that f : H → H is an LS -vector field, f(x) =Lx + K(x), L : H → H an isomorphism and φt : H → H is an LS -flowgenerated by f . Let N be an isolating neighborhood for φt and S := inv(N).Then the following equality holds true

(2.14) χ(hLS (S)) = degL(f,N).

Proof. Let hLS (S) = (En, εn)n≥n(E) and assume that n0 is chosen such thatχ(hLS (S)) = (−1)ρ(n)χ(En) (cf. Remark (2.12)) and

degLS (I + L−1K,N) = deg(I + PnL−1K,Nn)

for all n ≥ n0. According to the finite dimensional formula (1.18) one has

(−1)ρ(n)χ(En) = (−1)ρ(n) deg(L+ PnK,Nn).

Thus

χ(hLS (S)) = (−1)ρ(n)χ(En) = (−1)ρ(n) deg(L+ PnK,Nn)

= (−1)ρ(n) degL|Hn · deg(I + PnL−1K,Nn)

= deg(I + PnL−1K,Nn) = degLS (I + L−1K,N) = degL(f,N),

since the degree of the linear isomorphism L|Hn with respect to 0 is (−1)ν ,where ν is the number of negative eigenvalues of L. But in this case it isexactly dimHn

− =∑ni=1 dimH−i =

∑n−1i=0 ν(i) = ρ(n). This completes the

proof.

Now consider a weaker assumption about the operator L : H → H. Wewould like to admit the case when L is not an invertible operator, but isselfadjoint, i.e., 〈Lx, y〉 = 〈x, Ly〉 for all x, y ∈ H. Let P0 : H → H denotethe orthogonal projection onto H0, the kernel of L. Define L : H → H byLx := Lx + P0x. Since the kernel of L is orthogonal to the image of L, wesee that L is an isomorphism. In particular, if L is invertible, then L = L. Iff is a vector filed of the form Lx+K(x), where K is completely continuous,we can write it equivalently as

f(x) = Lx+ K(x),

where K(x) = K(x)− P0x. Note that K is completely continuous as well,since dimH0 < ∞. As before for an open bounded subset U ⊂ H andLS -vector field f = L+K, such that 0 6∈ f(∂U), we set

degL(f, U) := degLS (I + L−1K, U).

(2.15) Theorem. Assume that f : H → H is an LS -vector field with aselfadjoint linear part L : H → H and φt : H → H is an LS -flow generatedby f . Let N be an isolating neighborhood for φt and S := inv(N). Then

(2.16) χ(hLS (S)) = degL(f,N).


Proof. If L is selfadjoint then

deg(L+ PnK,Nn) = deg(L+ PnK,N

n),

since PnP0 = P0 and L preserves the splitting of H =⊕∞

n=1 Hn. Next

deg(L+ PnK,Nn) = deg L|Hn · deg(I + PnL

−1K,Nn).

Observe that deg L|Hn = (−1)ρ(n). Indeed, the number of negative eigen-values of L and L coincide, because L differs from L only on the kernel ofL by the identity. That is there are only the λ = 1 of multiplicity dimH0

added to spectrum of L. The deg(I + PnL−1K,Nn) stabilizes for large n

and represents degLS (I + L−1K,N) = degL(f,N). The result follows by(2.14).

In fact, this theorem can be formulated for much larger class of operatorsL. It is easy to see that L is admissible ifH = KerL⊕imL, where ⊕ denotesa direct sum (not orthogonal). This condition allows us to define the degLin the above way.

As a corollary we are going to formulate some properties of the numbersχ(hLS (S)) which are immediate consequence of the properties of the Leray–Schauder degree.

(2.17) Corollary. Suppose S is an isolated invariant set of an LS -flowgenerated by an LS -vector field f . Then the number χ(hLS (S)) has thefollowing properties:

Existence If χ(hLS (S)) 6= 0, then S contains a rest point of the flow;

Additivity If S1, S2 are the isolated invariant subsets of S and all ze-ros of f are contained in S1 ∪ S2, then χ(hLS (S)) = χ(hLS (S1)) +χ(hLS (S2));

Homotopy invariance Let U ⊂ H be an open set and N1, N2 be a boundedclosed sets contained in U . If S1 = inv(N1) and S2 = inv(N2) areisolated invariant sets of the LS -flows generated by the homotopic1

LS -vector fields, then χ(hLS (S ′)) = χ(hLS (S ′′)).

As it was pointed out by McCord in [33], the additivity property followsfrom the Morse inequalities [23]. Although, while the Morse inequalitiesgive us more information about the dynamics, the additivity in the aboveCorollary does not assume that the collection of sets S1 and S2 fulfills theadmissibility condition (admissible ordering), which is essential in the Morseinequalities approach.

1It is understood that the homotopy is assumed to be admissible from the degreetheory point of view. That is, the LS -vector fields f0 and f1 are homotopic if thereexists a homotopy ht connecting f1 and f2 and such that ht(x) 6= 0 for x ∈ ∂U and allt ∈ [0, 1].


2.2.1 Alternative approach in the particular case. In this sectionthe equality (2.14) will be obtained via direct calculation, in the case whenL = (−I, I) : H−⊕H+ → H−⊕H+ and S being an isolated zero of a givenvector field.

(2.18) Definition. We say, that a sequence Pn∞n=1, Pn : H → H is stronglyconvergent to the identity I : H → H, if lim

n→∞Pnx = x for all x ∈ H.

(2.19) Lemma. If K : H → H is a compact operator and Pn : H → H,n = 1, 2, . . . is a sequence of orthogonal projections onto Hn that is stronglyconvergent to the identity, then

(1) limn→∞

‖PnK −K‖ = 0;

(2) limn→∞

‖PnKPn −K‖ = 0;

(3) limn→∞

‖QnK‖ = 0, where Qn : H → H is the orthogonal projection ontoHn.

Proof. Statement (1) is a well known from the Riesz–Schauder theory. Since

‖PnKPn −K‖ ≤ ‖PnKPn − PnK‖+ ‖PnK −K‖

and since PnK is compact, in order to prove (2) it is enough to show thatfor any compact A one has lim

n‖APn − A‖ = 0. If A is compact, then the

adjoint operator A∗ is compact as well and we may write ‖APn − A‖ =‖(APn − A)∗‖ = ‖PnA∗ − A∗‖ → 0. Finally, we have an estimation

0 ≤ ‖QnK‖ ≤∥∥∥(∑∞

i=nQi

)K∥∥∥ = ‖(I − Pn−1)K‖ < ε

provided n ≥ n0. This proves (3).

(2.20) Definition. We say that A ∈ B(H) is hyperbolic, if

dist(σ(A), iR) := infλ∈σ(A), x∈iR

|x− λ| > 0.

The set of all hyperbolic operators will be denoted by Bhip(H).

Recall, that the multivalued map B(H) 3 A 7→ σ(A) ⊂ C is upper semicontinuous, that is for all A ∈ B(H) and ε > 0, there exists δ > 0, suchthat inequality ‖A−B‖ < δ implies sup

λ∈σ(B)

dist(λ, σ(A)) < ε.

(2.21) Lemma. Bhip(H) is an open subset of B(H).

Proof. Set ρ := dist(σ(A), iR). There exists δ > 0 such that for all B inδ-neighborhood of A

supλ∈σ(B)

dist(λ, σ(A)) < ρ/2.


Thus, the triangle inequality gives us the following estimation

dist(σ(B), iR) = infµ∈σ(B), x∈iR

|µ− x| ≥ infµ∈σ(B), λ∈σ(A), x∈iR

(|x− λ| − |λ− µ|)

≥ infλ∈σ(A), x∈iR

|x− λ| − supµ∈σ(B)

( infλ∈σ(A)

|λ− µ|) > ρ− ρ

2=ρ

2> 0,

which completes the proof.

(2.22) Theorem. Assume that f(x) = Lx + K(x) is an LS -vector fieldon H and L : H → H is such that

〈Lx, x〉 = ‖x+‖2 − ‖x−‖2 ,

where x = (x−, x+) ∈ H− ⊕ H+, both H± are of infinite dimension. Letf(0) = 0, Df(0) ∈ Bhip(H) and φt is an LS -flow generated by f . ThenS = 0 is an isolated invariant set for φt and there exists ρ > 0, such that

(2.23) χ(hLS (S)) = degL(f,B%).

Here B% stands for the open ball in H of radius %

Proof. The assumption f(0) = 0 and Df(0) ∈ Bhip(H) guarantees thatS = 0 is an isolated invariant set and x0 = 0 is isolated in the set f−1(0)(cf. Remark 1.11 in [1]). In order to compute the index on the left-handside of (2.23) consider a sequence of finite dimensional approximations

fn : Hn → Hn, fn(x) = Lx+ PnK(x).

Let A := DK(0) and notice that A is a compact linear map. Since thederivative Df(0) = L+A is a hyperbolic operator, then by Lemmas (2.19)and (2.21) there exists n0 ∈ N such that Dfn(0) = L + PnA is hyperbolic,provided n ≥ n0.

The closure of Bn% := B% ∩ Hn is an isolating neighborhood for the

invariant set Sn := 0 ⊂ Hn for the flow φtn generated by fn for n ≥ n1

(cf. Lemma (2.7)). Assume that n0 is chosen such that n0 ≥ n1. One has asplitting Hn0 = Hn0

− ⊕Hn0+ where Hn0

− (resp. Hn0+ ) stands for unstable (resp.

stable) subspace of the linear equation x = −Dfn0(0)x. In the hyperboliccase, the Conley index is exactly the homotopy type of a pointed sphere:h(Sn0) = [Sdim H

n0− , ∗].

Denote by En0 the space that is homotopy equivalent to (Sdim Hn0− , ∗). In

order to establish the relation between En0 and En0+1, we have to computethe index of the flow generated by fn0+1 : Hn0+1 → Hn0+1. Note that thederivative Dfn0+1(0) = L+ Pn0+1A preserves the splitting Hn0 ⊕Hn0+1. Itis easily seen if we write it in the following way

L|Hn0 + Pn0A+ L|Hn0+1 +Qn0+1A : Hn0 ⊕Hn0+1 → Hn0 ⊕Hn0+1.


Hence φtn0+1 is a product flow and one has h(Sn0+1) = h(Sn0) ∧ h(0, η),where h(0, η) is an index of 0 ⊂ Hn0+1 with respect to the flow gener-ated by x = −L|Hn0+1x−Qn0+1Ax.

Since ‖QnDK(0)‖ → 0, the maps L|Hn0+1 and L|Hn0+1 + Qn0+1A arehomotopic for sufficiently large n0 and the index h(0, η) is determined bythe dimension of the unstable subspace of the linear equation

x = −L|Hn0+1x.

Set Hn0+1 = H−n0+1 ⊕ H+n0+1, where H

−n0+1 (resp. H+

n0+1) is the unstable(resp. stable) subspace of L and define ν : N∗ → N∗ by ν(n) = dimH−n+1.One has

h(Sn0+1) = h(Sn0) ∧ [Sν(n0), ∗] = [Sν(n0)Sdim Hn0− , ∗]

(2.24) Corollary. En+1 is the ν(n)-fold suspension of En, provided that nis sufficiently large.

Define ρ : N∗ → N∗ by ρ(0) = 0 and ρ(n) =∑n−1i=0 ν(i). According to

definition of cohomological Conley index one has an isomorphism

CHq(hLS (S)) ∼= Hq+ρ(n)(h(Sn)), n ≥ n0.

It follows that CHq(hLS (S)) ∼= Hq+ρ(n)(Sdim Hn− , ∗) ∼= Z for q = dim Hn

− −ρ(n) and hence

χ(hLS (S)) = (−1)dim Hn−−ρ(n), n ≥ n0.

In particular we have χ(hLS (S)) = (−1)dim Hn0− −ρ(n0). By the stability prop-

erty of the Leray–Schauder degree

degL(f,B%) = degLS (I + L−1K,B%) = deg(I + L−1PnK,Bn% ) for n ≥ n0.

From the fact that deg(L|Hn , Bn% ) = (−1)ρ(n) and

deg(L+ PnK,Bn% ) = (−1)dim Hn

−

we conclude that

deg(I + PnL−1K,Bn) = deg(L+ PnK,B

n% ) · [deg(L|Hn , Bn

% )]−1

= (−1)dim Hn−−ρ(n)

for n ≥ n0 and the proof of (2.23) is completed.

(2.25) Example. Let H = `2 with the standard basis (en)∞n=1 and decom-pose H as follows: H = H1 ⊕H2, where

H1 =x ∈ `2 : x2n−1 = 0, n ∈ N

, H2 =

x ∈ `2 : x2n = 0, n ∈ N

.


Let L(x, y) = (−x, y). It can be written as an infinite diagonal matrix

L =

−1 0 0 · · ·0 1 0 · · ·0 0 −1 · · ·...

...... . . .

Notice that both stable and unstable subspaces of L are infinite dimensional.Define two linear compact operators K1,2 : H → H by the formula

Ki (en) := (−1)i5

2nen.

Let B% := x ∈ H; ‖x‖ < % and let f : H → H be an LS -vector field suchthat f(x) = Lx+K1(x) inside the ball Br and f(x) = Lx+K2(x) outsidethe ball BR for some 0 < r < R <∞. It is easy to check that

degL(f,Br, 0) = degL(f,BR, 0) = −1.

The origin is an isolated invariant set of the flow induced by x = −f(x). Inorder to compute its LS -Conley index let us introduce the subspaces

Hn =x ∈ `2 : xi = 0, i 6= 2n− 1, 2n

.

Then each of these subspaces is two dimensional, Hn’s are mutually or-thogonal L-invariant and H = ⊕∞n=1Hn. Each Hn can be represented asH−n ⊕H+

n , where H−n (resp. H+n ) is repelling (attracting) subspace of L|Hn .

Here we have ν(n) = dimH−n+1 = 1. Set Si := invBi, i = r, R. One caneasily check that hLS (Sr) is a homotopy type of a spectrum E such thatEk = Sk+1, a pointed (k+ 1)-sphere for k ≥ 1 while hLS (SR) is a homotopytype of a spectrum E ′, where E ′k is a pointed sphere of dimension k− 1 fork ≥ 1. The computations of cohomology give us

Hq(E) =

Z, q = 1;0, else. Hq(E ′) =

Z, q = −1;0, else.

and the Euler characteristic of both hLS (Sr) and hLS (SR) is equal −1. Theconclusion is that Sr 6= SR and it can not be captured via the degree theory.

Chapter 3

Equivariant theory

So far we have investigated the homotopy invariants for flows and vectorfields having in mind a close relationship between them. We have estab-lished a formula relating the degree of a compact field f : H → H to theConley index of an isolated invariant set of a flow generated by f . Since thetopological degree as well as the Conley index have equivariant counterparts,the natural task is to investigate the relation joining these topological tools.Various definitions and properties of the equivariant degree have been inten-sively studied by many authors. Let us only mention the following: Dancer[8], Gęba, Krawcewicz, Wu [17], Ize, Massabo, Vignoli [21, 22] and Rybicki[43]. In this thesis we are going to deal with the degree for equivariant gra-dient mappings introduced by Gęba in [14]. Although the restriction to theclass of the potential maps is appreciable, this degree became a very pow-erful tool for the variational approach to many problems (cf. [18, 40, 44]).It is worth to noticing that this restriction (gradient mappings) does nottake any contributions when the symmetries of finite groups are taken intoconsideration. This is related to the result originally due to Parusiński [35]which says that gradient maps are homotopic iff they are gradient homo-topic. The Z2-equivariant version of this result is presented in the paper byJanczewska and the author [29].

The equivariant Conley index has been introduced by Floer in [12] inorder to study hyperbolic invariant sets and some bifurcation questionsfor Hamiltonian systems in the joint paper (Floer and Zehnder [13]). Anextension of this index to infinite dimensional spaces, motivated by [16],has been defined by Izydorek in [24]. This thesis is concerned, after all,only with the finite dimensional version of the G-index. Nevertheless theauthor is aware of the fact that the results contained herein may have anatural generalization to the infinite dimensional G-flows and equivariantvector (compact) fields. The work is in progress.

In the presence of a group action the role of the Euler characteristic of aG-space X plays the universal additive invariant u(X), the element of thering U(G) (called the Euler ring) associated with a compact Lie group G.The universal additive invariant has the same properties as the usual Eulernumber. In particular, if G is a trivial group, u(X) coincides with χ(X).In Subsection 3.1.4 we describe the ring U(G), and give some basic factsabout u(X).

In this chapter we will repeat some definitions that can be found in theprevious part of the thesis in order to put the known notions in the context of

3.1 Basic equivariant topology 26

G-equivariant theory. The main result of this part is the equivariant Morse–Conley–Zehnder equation in Section 3.4. This tool allows us to capture therelationship between equivariant Conley index and degree for equivariantgradient maps. Then, in Subsection 3.4.2, this equation is used to provesimple multiplicity results for critical orbits of an invariant function. Finallyin Section 3.5 we shall extend the Reineck continuation theorem to theequivariant setting.

3.1 Basic equivariant topology.

(3.1) Definition. A compact Lie group is a topological group G such that

• G has a structure of a smooth compact manifold;

• the composition map G×G→ G; (g, h) 7→ gh−1 is smooth.

Throughout this chapter G stands for a compact Lie group. One canalways think of G to be a finite group or a subgroup of Aut(RN) the groupof all linear isomorphism of RN .

(3.2) Definition. A linear representation V (G-representation) of a groupG is a pair (Rn, ρ) consisting of the Euclidean vector space and a homomor-phism

(3.3) ρ : G→ Aut(Rn).

The representation is called orthogonal if the image of (3.3) is containedin the group O(n) of all linear isomorphisms of Rn preserving the innerproduct, i.e., 〈ρ(g)x, ρ(g)y〉 = 〈x, y〉 for all g ∈ G and vectors x, y ∈ Rn.

Two linear representations V = (Rn, ρ) and W = (Rn, ς) are said to beisomorphic iff there exists an invertible linear mapping ` : V → W such thatfor g ∈ G

`ρ(g) = ς(g)`.

The following basic result can be found for example in [3].

(3.4) Theorem. Each linear G-representation is isomorphic to an orthog-onal representation.

It allows us to focus, without any loss of generality, our attention on theorthogonal G-representations. Unless we comment to the contrary, we shallalways assume that V is an orthogonal representation.

An action of a group G (G-action) on a topological space X is a (con-tinuous) map G×X → X sending (g, x) to gx and satisfying the followingproperties:

• ex = x for all x ∈ X, where e stands for the identity of G;

• h(gx) = (hg)x for all g, h ∈ G and x ∈ X.


A G-space is a pair consisting of an underlying space X with a given G-action.

If V is a G-representation there is a natural linear action of a group Gon Rn given by (g, x) 7→ ρ(g)x. For an abbreviation we will write gx insteadof ρ(g)x.

A subset Ω of a G-space X is called a G-invariant (a G-set) providedthat x ∈ Ω and g ∈ G imply gx ∈ Ω. If X and Y are G-spaces, then acontinuous map f : X → Y is called a G-equivariant map (a G-map) if therelation f(gx) = gf(x) holds for all x ∈ X and g ∈ G.

3.1.1 Orbit types. LetG be a compact Lie group. The subgroupH ⊂ Gis called conjugate to a subgroup K ⊂ G if there is g ∈ G such thatH = g−1Kg. The conjugacy defines an equivalence relation, and we willwrite (H) for a conjugacy class of H. The set of all conjugacy classes ofclosed subgroups of G will be denoted by Φ(G). The set Φ(G) is partiallyordered. We write

(3.5) (H) ≤ (K) if gHg−1 ⊂ K for some g ∈ G.

Assume that X is a G-space. The isotropy group at a point x ∈ X is asubgroup of G defined by

Gx := g ∈ G; gx = x .

That is, this is the set of those elements of G that leave point x fixed.The isotropy group at x measures somehow the symmetry of point x. Themost symmetric are those, whose isotropy group is the whole G, e.g. theorigin of a representation. For each x ∈ X, the group Gx is closed in G.Given A ⊂ X, the G-orbit of A is defined by GA = ga; g ∈ G, a ∈ A. Inparticular the set Gx = gx; g ∈ G is the G-orbit through x (G-orbit ofx). Isotropy groups of points on the same G-orbit are conjugate subgroupsof G, precisely Ggx = g−1Gxg.

(3.6) Definition. Points x, y ∈ X have the same orbit type if Gx and Gy

are conjugate subgroups of G.

Hence the points on the same orbit have the same orbit type. Since theorbit type is determined by a conjugacy class, the set Φ(G) of all conjugacyclasses will be called the set of orbit types1.

(3.7) Theorem ([30], Corollary 4.25). The set of orbit types of a finitedimensional representation of a Lie group is finite.

1In [3], an orbit type is called an isotropy type. The notion of orbit type is used forequivalence class of G-orbits under equivariant homeomorphism. However each orbit typecontains a coset space G/H and the type of G/K equals the type G/K iff (H) = (K).The main difference lies in the partial ordering of the set of orbit types. If P ' G/Hand Q ' G/K then type(P ) ≥ type(Q) iff (H) ≤ (K).


Let H be a closed subgroup of G. Throughout the rest of this chapterwe will use the following notation:

XH = x ∈ X; H ⊂ Gx = x ∈ X; hx = x, h ∈ HX(H) = GXH = x ∈ X; (H) = (K) forK ⊂ GxXH = x ∈ X; Gx = HX(H) = GXH = x ∈ X; (Gx) = (H)X>(H) = X(H) \X(H) =

⋃(K)>(H)

X(K).

(3.8) Definition. An orbit Gx and its orbit type (Gx) are called principalif Gx has a G-invariant open neighborhood that contains no orbit of smallerorbit type with respect to the partial order (3.5).

The significance of the notion of principal orbits asserts the following

(3.9) Theorem ([3], Theorem 3.1). Let (H) be a principal orbit type. Thenthe union V(H) of orbits of principal type is open and dense in V .

If V is a G-representation, then V H is a linear subspace of V , not neces-sarily a G-subrepresentation. Although on V H we have an induced actionof the normalizer NH := g ∈ G; g−1Hg = H. Indeed for x ∈ V H andn ∈ NH one has nx = nhx = h′nx for some h, h′ ∈ H. Hence nx is fixedby the action of H.

If H is a closed subgroup of a compact Lie group G, then there is anatural action of G on the left coset space G/H by left translations, i.e.,G×G/H → G/H, (g′, gH) 7→ g′gH. It is a typical example of a homogenousspace. Moreover it is a classical result that G/H is a smooth compact G-manifold (cf. for instance [30]).

The next proposition is a statement of the fundamental facts concerningactions of a compact Lie groups on G-manifolds.

(3.10) Proposition ([11], p. 63). Let M be a G-manifold. Then

• G-orbits are compact G-invariant submanifolds of M ;

• if x ∈ M , Gx is equivariantly diffeomorphic to G/Gx via the mapsending gx to the coset gGx;

• if G acts freely on M , then M/G has a unique smooth structure withrespect to which the canonical surjection p : M →M/G is the principalG-bundle.

3.1.2 Slices. Given a closed subgroup H ⊂ G and a H-space X one candefine an action H ×G×X → G×X setting (h, (g, x)) 7→ (gh−1, gx). Theorbit space of this action is denoted by G×HX and is called twisted productof G and X. For an abbreviation we denote the equivalence classes just byits representatives (g, x).


The space G×HX carries an action of a group G defined by (g′, (g, x)) 7→(g′g, x). If f : X → Y is a H-map, there exists an induced G-map

G×H f : G×H X → G×H Y

given by (G ×H f)(g, x) := (g, f(x)). If V is an H-representation one canshow that G×H V is a differentiable manifold and the projection G×H V

π−→G/H is a differentiable G-fibre bundle with π−1(gH) ∼= V (that is, is a G-vector bundle). For details, see [10]. Recall that the surjection π : E →M isa G-vector bundle if E andM are G-spaces, π is a G-map and g : Ex → Egxis a linear isomorphism. Of course the condition of local triviality is needed.

(3.11) Theorem (The Slice theorem, [30] p. 184). Let G be a compact Liegroup and M a smooth G-manifold. For every x ∈ M , the orbit Gx is aG-invariant submanifold of M . Let N denote the normal G-vector bundleof Gx in M . Then the fibre Nx over x of N is a representation space of theisotropy group Gx so that N is isomorphic to

G×Gx Nx → G/Gx

as a smooth G-vector bundles. Moreover there exists a G-invariant openneighborhood U of Gx in M and a G-diffeomorphism f : G ×Gx Nx → Usuch that the restriction of f to the zero section gives the G-diffeomorphismfrom G/Gx onto Gx. If we take the G-invariant Riemannian metric onM then the Gx-action on Nx is given by the Gx-action on the orthogonalcomplement of TxGx in TxM .

3.1.3 G-complexes. The object of our interest, the Conley index, is ahomotopy type of a pointed space which supports the structure of CW-complex. Although the notion of CW-complex is well known in topology,we present here some basic definitions, since the G-equivariant Conley indexjoins the notion of CW-complex and G-space. The definitions are borrowedfrom the paper by Gęba and Rybicki [15].

We use the standard notation Sn−1 = x ∈ Rn; ‖x‖ = 1 and Dn =x ∈ Rn; ‖x‖ ≤ 1 for the unit (n − 1)-sphere and the unit n-ball in Rn

respectively. In what follows we assume that Dn carries the trivial G-action,i.e., gx = x for all x ∈ Dn and g ∈ G. We set Bn = Dn \ Sn−1.

(3.12) Definition. Let (X,A) be a compact pair of G-spaces and Hj,j = 1, 2, . . . , q be a family of closed subgroups of G. We say that X isobtained from A by simultaneously attaching the family of equivariant k-cells of orbit type (Hj); j = 1, . . . , q if there exists a G-map

ϕ :q⊔j=1

Dk ×G/Hj → X

which mapsq⊔j=1

Bk ×G/Hj homeomorphically onto X \A. We call ϕ(Dk ×

G/Hj) a closed k-dimensional cell of orbit type (Hj).


(3.13) Definition. Let X be a compact G-space. A finite equivariant CW-decomposition of X consists of an increasing family of G-subsetsX0 ⊂ X1 ⊂. . . ⊂ Xn = X and a family

n⋃k=0Hj,k; j = 1, . . . , q(k) of closed subgroups

of G such that

• X0 =q(0)⊔j=1

G/Hj,0;

• the space Xk is obtained from Xk−1 by simultaneously attaching thefamily of equivariant k-cells of orbit type (Hj,k); j = 1, . . . , q(k) foreach 1 ≤ k ≤ n.

A pointed G-space is a pair (X, x0) where X is a G-space with a dis-tinguished point x0 called the base point and such that the action of Gleaves the base point fixed. The pointed G-spaces are the objects of thecategory whose morphisms are G-maps preserving the base point. If X is aG-space without base point, then the superscript plus X+ means that X isconsidered as a pointed space with a separate base point added.

(3.14) Definition. Let (X, x0) be a pointed compact G-space. A pointedfinite equivariant CW-decomposition of (X, x0) consists of an increasingfamily of G-subsets X−1 ⊂ X0 ⊂ X1 ⊂ . . . ⊂ Xn = X and a familyn⋃k=0Hj,k; j = 1, . . . , q(k) of closed subgroups of G such that

• X−1 = x0;

• X0 = x0 tq(0)⊔j=1

G/Hj,0;

• the space Xk is obtained from Xk−1 by simultaneously attaching thefamily of equivariant k-cells of orbit type (Hj,k); j = 1, . . . , q(k) foreach 1 ≤ k ≤ n.

The familyn⋃k=0Hj,k; j = 1, . . . , q(k) is called the orbit type of the de-

composition of X. For short we use the term G-complex (pointed G-complex) for a (pointed) G-space if there exists a (pointed) finite equivariantCW-decomposition of X (resp. (X, x0)).

3.1.4 Euler ring U(G). If (X, x0) and (Y, y0) are pointed G-spaces(gx0 = x0 and gy0 = y0 for all g ∈ G) then we say that (X, x0) and(Y, y0) have the same G-homotopy type iff there exist a pair of G-mapsf : (X, x0) → (Y, y0) and g : (Y, y0) → (X, x0) such that gf ∼G id(X,x0) andfg ∼G id(Y,y0). The symbol ∼G means that if Ht is a homotopy joining twoG-equivariant maps, then for all t ∈ [0, 1] the map Ht is a G-map as well.Of course, the relation ∼G is an equivalence and the equivalence class underrelation ∼G is denoted by [X]G. We say that [X]G is the G-homotopy typeof X.


Let us introduce the symbol F (G) for the category whose objects arepointed G-complexes and F [G] for the set of all G-homotopy types ofpointed G-complexes. For (X, x0), (Y, y0) ∈ F (G) we define its wedge sumto be

X ∨ Y := (X × y0 ∪ x0 × Y, (x0, y0)) ∈ F (G).

and its smash product

X ∧ Y := X × Y/X ∨ Y.

Of course we also have X ∧ Y ∈ F (G).Let F = Z[F [G]] be the free abelian group generated by the G-homotopy

classes of pointed G-complexes and let N be the subgroup of F generated byall elements [A]G−[X]G+[X/A]G, where A is a pointed G-subcomplex of X.Define U(G) := F/N. The class of [X]G ∈ F [G] under this identificationwill be denoted by u(X). Directly from the definition of U(G) we see thatthe addition can be obtained via the wedge sum

u(X) + u(Y ) = u(X ∨ Y ).

Moreover the assignment (X, Y ) 7→ X ∧ Y induces the multiplication inU(G) (cf. [10]), that is

u(X)u(Y ) = u(X ∧ Y ).

(3.15) Definition. The set U(G) with the composition laws defined asabove is called the Euler ring of the group G.

As we have mentioned earlier the coset spaceG/H of a compact Lie groupover the closed subgroup H is a smooth compact G-manifold and hence, dueto theorem of Illman in [20], is a G-complex. Therefore G/H+ ∈ F (G) andwe can consider the element u(G/H+) ∈ U(G). In what follows we willwrite uG(H) instead of u(G/H+). The abelian group structure of the ringU(G) is fairly easy and its description is given in the following statement.

(3.16) Proposition ([10]). As a group U(G) is the free abelian group withbasis uG(H), where (H) ∈ Φ(G). If X ∈ F (G), then

(3.17) u(X) =∑

(H)∈Φ(G)

χ(X(H)/G,X>(H)/G)uG(H).

Here χ stands for the Euler characteristic of the pair of CW-complexes. Asa ring U(G) is commutative with the unit uG(G).

The Euler characteristic of a cell complex K can be expressed as analternating sum χ(K) =

∞∑k=0

(−1)ksk, where sk is the number of k-cells in

the complex K. This formula holds in an equivariant setting as well and wehave a nice tool for computations.

3.2 Degree for G-equivariant gradient maps 32

(3.18) Proposition ([15]). Let X ∈ F (G) and letn⋃k=0Hj,k; j = 1, . . . , q(k)

be an orbit type of the decomposition of X. Then

(3.19) u(X) =∑

(H)∈Φ(G)

n(H)(X)uG(H),

where n(H)(X) =n∑k=0

(−1)kν((H), k) and ν((H), k) is the number of equiv-

ariant k-cells of orbit type (H).

3.2 Degree for G-equivariant gradient maps.

In this section we briefly recall definition of the degree for gradient G-maps presented in [14]. Paper [14] is the main reference for this section,where the reader can find proofs of theorems discussed below.

We say that a function ϕ : V → R is G-invariant if ϕ is constant on theorbits of G, i.e., ϕ(gx) = f(x) for x ∈ V and g ∈ G. If f : V → V is agradient of some continuously differentiable G-invariant function f = ∇ϕ,then we call it G-equivariant gradient map. As an immediate consequenceof the above definition and the chain rule we get the property that f(gx) =gf(x) for all x ∈ V and g ∈ G.

In the same manner we define the homotopy joining two equivariantgradient maps. That is the map h : V × [0, 1] → V is a gradient G-homotopy if there exists a G-invariant function q : V × [0, 1] → R of classC1 (q(gx, t) = gq(x, t)) such that h(x, t) = ∇q(x, t) for all t ∈ [0, 1]. Thegradient is taken with respect to the x variable.

Fix an open bounded and G-invariant subset Ω ⊂ V and f : V → V agradient G-map. We say that a pair (f,Ω) is ∇G-admissible provided thatf(x) 6= 0 for x ∈ ∂Ω. In other words an ∇G-admissible pair is an equivariantmap of pairs f : (V, ∂Ω)→ (V, V \0). We say that two∇G-admissible pairs(f0,Ω) and (f1,Ω) are ∇G-homotopic if there exists a gradient ∇G-homotopyh : V × [0, 1]→ V connecting them, i.e., hi = fi, i = 0, 1 and such that thepair (ht,Ω) is ∇G-admissible for t ∈ [0, 1].

From now on f : V → V will always mean an equivariant gradient map.Let x be a fixed point in V of an orbit type (H), i.e., H = Gx. We have anorthogonal splitting

(3.20) V = Tx(Gx)⊕Wx ⊕Nx,

where Wx is the orthogonal complement of Tx(Gx) in the tangent spaceTx(V(H)) and Nx = Tx(V(H))

⊥. Assume that x ∈ f−1(0) and f is differen-tiable at x. Then Tx(Gx) ⊂ KerDf(x) since for all g ∈ G f(gx) = 0. Noticethat Tx(V(H)) is an invariant subspace of the linear map Df(x). Thereforewith respect to the decomposition (3.20) Df(x) has a form

(3.21)

0 0 00 Kf(x) 00 0 Lf(x)

,

3.2 Degree for G-equivariant gradient maps 33

i.e., Kf(x) := Df(x)|Wx and Lf(x) := Df(x)|Nx .

(3.22) Definition. An orbit Gx is called a regular zero orbit of f , iff(x) = 0 and KerDf(x) = Tx(Gx). It means that the map Kf(x) ⊕Lf(x) : Wx ⊕ Nx → Wx ⊕ Nx is an isomorphism. The Morse index of theregular zero orbit Gx is defined to be the number of negative eigenvalues ofKf(x), k := dimW−

x . We set σ(Gx) := (−1)k.

For an open G-set U such that U is a compact subset of V(H) and ε > 0define

N(U, ε) := v ∈ V : v = x+ n, x ∈ U, n ∈ Nx, |n| < ε .

The set N(U, ε) will be called a tubular neighborhood of type (H) providedthat the decomposition v = x + n is unique. Let N(U, ε) be a tubularneighborhood of type (H). The gradient equivariant map f is (H)-normalon N(U, ε) if for all v = x+ n ∈ N(U, ε)

f(v) = f(x) + n.

(3.23) Definition (Generic pair). We say that ∇G-admissible pair (f,Ω) isgeneric if there exists an open G-subset Ω0 ⊂ Ω such that

(Gen.1) f−1(0) ∩ Ω ⊂ Ω0;

(Gen.2) f|Ω0 is of class C1;

(Gen.3) f−1(0) ∩ Ω0 is composed of regular zero orbits;

(Gen.4) for each H with Z = f−1(0) ∩ Ω(H) 6= ∅ there exists a tubularneighborhood N(U, ε) of type (H) such that Z ⊂ N(U, ε) ⊂ Ω and fis (H)-normal on N(U, ε).

The next theorem allows us to define the gradient degree for a ∇G-admissible pair (f,Ω).

(3.24) Theorem (Generic Approximation Theorem, [14]). For any ∇G-admissible pair (f,Ω) there exists a generic pair (f1,Ω) such that (f,Ω) and(f1,Ω) are ∇G-homotopic.

(3.25) Lemma ([14]). If (f,Ω) is a ∇G-admissible pair then there exists agradient G-map f1 : V → V such that (i) f1(x) = f(x) for x ∈ V \ Ω and(ii) (f1,Ω) is ∇G-admissible and generic.

Proof of Theorem (3.24). Let (f1,Ω) is ∇G-admissible and generic pair fromthe Lemma (3.25). Define G-homotopy h : V × [0, 1] → V as h(x, t) =(1 − t)f(x) + tf1(x). Clearly the pair (h( · , t),Ω) is ∇G-admissible for allt ∈ [0, 1].

3.3 Equivariant Conley index 34

(3.26) Definition. Let (f,Ω) be a ∇G-admissible pair. The G-equivariantgradient degree of (f,Ω) is an element of the Euler ring U(G) defined as

deg∇G(f,Ω) :=∑

(H)∈Φ(G)

n(H)uG(H),

wheren(H) :=

∑(Gxi )=(H)

σ(Gxi)

and Gxi are the disjoint orbits of type (H) in f−11 (0) ∩ Ω. Here (f1,Ω) is

any generic pair G-homotopic to (f,Ω).

The above definition is correct because of the following

(3.27) Theorem ([14]). If two generic pairs (f0,Ω) and (f1,Ω) are G-homotopic then

deg∇G(f0,Ω) = deg∇G(f1,Ω).

(3.28) Example. Let Ω = (x, y) ∈ R2; 1/2 < x2 + y2 < 3/2 and the ac-tion of G = SO(2) on the real plane is given simply by rotation, i.e., for

(3.29) γθ =

[cos θ − sin θsin θ cos θ

]∈ SO(2), θ ∈ [0, 2π)

γθ(x, y) = (x cos kθ − y sin kθ, x sin kθ + y cos kθ), k ∈ N.

Hence V is a plane R2 with rotations by the angle kθ. Define ϕ : V → Rby the formula: ϕ(x, y) = −(x2 + y2 − 1)2. It is easy to check that ϕ isSO(2)-invariant function, and the pair (∇ϕ,Ω) is a ∇G-admissible. Eachpoint except the origin has an orbit type (Zk), that is V(Zk) = R2 \ (0, 0).The map f = ∇ϕ vanishes at the point (x0, y0) = (1, 0), and consequentlythe whole orbit G(1, 0) ≈ S1 is the set of zeros of f . The derivative at(1, 0) is a map (u, v) 7→ (−8u, 0) with the kernel KerDf(1, 0) = span([0, 1])which is exactly the tangent space T(1,0)G(1, 0). The Morse index of G(1, 0)is 1 and hence σ(G(1, 0)) = −1. Directly from the definition one obtainsdeg∇G(f,Ω) = −uG(Zk).

3.3 Equivariant Conley index.

As we have noticed earlier (cf. Example (1.5)) with a locally Lipschitzvector field v : Rn → Rn one can associate a local flow by integration of adifferential equation. More precisely, through each point x ∈ V passes amaximal integral curve φx : (αx, βx)→ Rn satisfying

dφxdt

(t) = v(φx(t))

φx(0) = x.


Setting D := (t, x) ∈ R× V ; t ∈ (αx, βx) and φ(t, x) := φx(t) we obtain alocal flow on V , that is (i) D ⊂ R× V is an open neighborhood of 0× Vand φ : D → Rn is continuous (ii) if (t, x) ∈ D and (s, φ(t, x)) ∈ D then(s+ t, x) ∈ D and φ(s, φ(t, x)) = φ(s+ t, x) (iii) φ(0, x) = x.

Now we are concerned with an equivariant vector fields. It is not sur-prising that the G-equivariant vector fields generate local G-flows. We aregoing to formulate this fact in the following

(3.30) Lemma (cf. Lemma 2.10.5 in [11]). Let V be a representation ofa compact Lie group G and v : V → V be a G-equivariant, locally Lipschizvector field. Then the differential equation

x(t) = v(x(t))

defines a local G-flow. That is (i) the set D ⊂ R × V is a G-set, i.e., if(t, x) ∈ D then (t, gx) ∈ D for all g ∈ G; (ii) φ(t, gx) = gφ(t, x) for all(t, x) ∈ D and g ∈ G.

Proof. The invariance of D under an action of a group comes from theinvariance of V . Let φx : (αx, βx)→ V represents an integral curve passingthrough a point x. Define φ := gφx : (αx, βx)→ V . Since

˙φ(t) = gφx(t) = gv(φx(t)) = v(gφx(t)) = v(φ(t))

and φ(0) = gx, then φ is a curve through a point gx. Hence (αx, βx) ⊂(αgx, βgx) and from the uniqueness of the solution φgx = gφx. Replacing inthe above argument x by gx and φ by g−1φgx we obtain (αgx, βgx) ⊂ (αx, βx)and φgx = gφx for all x ∈ G and x ∈ Ω. This show that φ is a G-flow.

From now on we will consider local flows generated by vector fields atleast of class C1. Without loss of generality we can assume that differentialequation x = v(x) generates a flow, i.e., the set D = R× V .

We repeat basic definitions and notions which are necessary for the def-inition of the Conley index in the presence of an action of a Lie group G.Let φ be a G-flow on V . For a G-set X ⊂ V the maximal invariant subsetunder the flow φ in X is given by

inv(X) =x ∈ X; φt(x) ∈ X, for all t ∈ R

.

Since X is G-invariant so is inv(X). If X is in addition compact andinv(X) ⊂ intX, then X is called an isolating neighborhood and inv(X)is an isolated invariant set. For an isolated invariant set there exists a G-index pair (N,L), i.e., the pair of compact G-invariant subsets of V suchthat (i) the closure of N \L is an isolating neighborhood; (ii) L is positivelyinvariant rel. N and (iii) if x ∈ N and φ[0,t](x) 6⊂ N for some t > 0, thenφs(x) ∈ L for some s ∈ [0, t]. For the existence of a G-index pair we referto [12, 13, 14].


The G-homotopy type of the quotient N/L does not depend on theparticular choice of the index pair. Recall that N/L is obtained from N bycollapsing all points in L to the point [L] which is distinguished in N/L.The action of G on N/L is induced from the action on N and g[L] = [L]for all g ∈ G.

Assume X ⊂ V is an isolated neighborhood of a flow φ.

(3.31) Definition. The G-equivariant Conley index of S := inv(X) de-noted by hG(S) (or sometimes hG(X,φ), to indicate the isolating neigh-borhood and the flow) is defined to be a G-homotopy type of a pointedG-space N/L, where (N,L) is an arbitrary G-index pair for S. That ishG(X,φ) := [N/L]G

The equivariant Conley index has the same properties as the ordinaryone. In particular the continuation property holds. We say that φ : R ×V × [0, 1] → V is a continuous family of G-flows on V if φλ : R × V → Vis a G-flow on V for all λ ∈ [0, 1], where φλ(t, x) = φ(t, x, λ). Notice thatwe do not restrict the class of flows to the gradient one if it is not specifiedotherwise.

(3.32) Proposition. Suppose that X is a compact G-subset of V and φ isa continuous family of G-flows on V . If X is an isolating neighborhood forφλ, λ ∈ [0, 1], then hG(X,φ0) = hG(X,φ1).

The next proposition asserts that the G-equivariant Conley index carriesthe structure of a G-complex (recall, that it means a finite G-CW-complex).In fact, the statement can be made stronger by eliminating the assumptionthat the vector field is a gradient, see Corollary (3.79). We will prove thisusing the continuation theorem for equivariant flows (cf. Theorem (3.74)).

(3.33) Proposition (Proposition 5.6. in [14]). Let (f,Ω) be a∇G-admissiblepair and let φt denote the G-flow generated by −f . Assume that Ω is anisolating neighborhood. Then G-index hG(Ω, φt) is a homotopy type of afinite G-complex

(3.34) Example. Let us reexamine the example (3.28) with G = SO(2).Consider the negative gradient G-flow on V = R2 given by

−∇ϕ(x, y) = (4x(x2 + y2 − 1), 4y(x2 + y2 − 1)).

The set N = Ω = (x, y) ∈ V ; 1/2 ≤ x2 + y2 ≤ 3/2 is an isolating G-invariant neighborhood and inv(N) = (x, y) ∈ V ; x2 + y2 = 1. The indexpair can be chosen to be (N, ∂N). The Conley index is a G-homotopy typeof a G-complex consisting of one 0-cell of orbit type (G) (as a distinguishedpoint with the trivial action) and one 1-cell of orbit type (Zk). Accordingto formula (3.19) we have u(hG(N)) = −uG(Zk). Notice that we do not takeinto account the distinguished point. Notice also, that u(hG(N)) coincidewith deg∇G(f,Ω) computed in Example (3.28).

3.4 Equivariant Morse–Conley–Zehnder equation 37

3.4 Equivariant Morse–Conley–Zehnder equation.

The Morse–Conley–Zehnder equation establishes the relationship be-tween the Conley index of an isolated invariant set with the indices of itsMorse decomposition, cf. Theorem (1.8). In particular, if the flow is givenby the gradient of a Morse function it generalizes the classical Morse in-equalities which give the estimation of the number of critical points by thetopological invariants of the underlying domain. In this section we shallgeneralize this result to the G-equivariant setting. We will use the equa-tion to derive some multiplicity results for critical G-orbits and to obtainthe relationship between the G-equivariant Conley index and the gradientequivariant degree.

The nonequivariant equation is expressed in terms of Poincaré polynomi-als with integer coefficients being the Betti numbers of certain index pairs.We are going to define the Betti numbers of G-equivariant Conley index andthen Poincaré polynomial appropriate for our purposes. Of course, one canexpect that in the case G = e (the trivial group), the obtained equationwill coincide with the classical one. Now and subsequently let H∗ denotethe Aleksander-Spanier cohomology with coefficients in some principal idealdomain R. This particular cohomology theory is chosen because it satis-fies the following strong excision property : Given two closed pairs (X,A)and (Y,B) in V and a closed continuous map f : (X,A) → (Y,B) suchthat f induces a bijection of X \ A onto Y \ B one has an isomorphismf ∗ : Hq(Y,B;R)→ Hq(X,A;R) for all q ≥ 0. For a more general statementof this fact we refer to the book by Spanier [46], Theorem 6.6.5.

It is worthwhile to mention that in our approach to the equivariant theorythere is no equivariant cohomology at all.

If E is a R-module then we set

rankE = dim(E ⊗R QR),

whenever dim(E ⊗R QR) is finite. Otherwise rankE =∞. Here QR standsfor the field of quotients of the ring R. The comparison of the classicalEuler characteristic with its equivariant analogue u(X) (defined merely fora homotopy type of a G-complexes, see Proposition (3.16)) being an el-ement of the Euler ring U(G) leads us to the conclusion that the k-thBetti numbers of X ∈ F (G) schould be the collection of the numbersrankHk(X(H)/G,X>(H)/G), where (H) ∈ Φ(G). Since we are concernedwith the G-index which is determined by an arbitrary G-invariant indexpair the following definition seems to be reasonable.

(3.35) Definition. Let (X,A) be a compact pair of G-invariant subsets ofV . The numbers

βq(H)(X,A) := rankHq(X(H)/G, (X>(H) ∪ A(H))/G), (H) ∈ Φ(G)

are called the q-th Betti numbers of the pair (X,A).


For an abbreviation we put

(XHA) := (X(H)/G, (X>(H) ∪ A(H))/G).

Recall that for a compact sets X ⊃ Y ⊃ Z there exists a connectinghomomorphism δq : Hq(Y, Z) → Hq+1(X, Y ), and a long exact sequence ofa triple (X, Y, Z):

(3.36) . . .δq−1

−−→ Hq(X, Y )ıq−→ Hq(X,Z)

q−→ Hq(Y, Z)δq−→ . . . ,

where ıq i q are homomorphisms induced by inclusions ı : (X,Z) → (X, Y )and : (Y, Z) → (X,Z) respectively. In order to overcome difficultiesconnected with definition of the Betti numbers we will need a couple oftechnical results.

(3.37) Lemma. If N2 ⊃ N1 ⊃ N0 is a triple of compact G-sets, (H) ∈Φ(G), then

H∗

N (H)1 /G,

N>(H)1 ∪N (H)

0

G

∼= H∗

N>(H)2 ∪N (H)

1

G,N>(H)2 ∪N (H)

0

G

.Proof. We are going to use the strong excision property of the Alexander-Spanier cohomology. Firstly we check that the pairs in question are closed.Indeed, for a closed G-subset N ⊂ V and a closed subgroup H ⊂ G one hasNH = N ∩ V H . Since V H is a linear subspace of V then V H is closed andso is NH . Further N (H) = GNH is closed, because the action of a compactLie group is a closed map (Theorem 1.1.2 in [3]). The set of orbit types ofa finite dimensional representation is always finite, hence N>(H) is closed asa finite sum of closed sets. Lastly, the set of orbits N/G endowed with thequotient topology is closed since the projection N → N/G taking x into itsorbit is closed (Theorem 1.3.1 in [3]). Clearly, the inclusion

e :

N (H)1 /G,

N>(H)1 ∪N (H)

0

G

→

N>(H)2 ∪N (H)

1

G,N>(H)2 ∪N (H)

0

G

is continuous and closed. Moreover for each x ∈ (N

(H)1 /G) \ ((N

>(H)1 ∪

N(H)0 )/G) one has e(x) = x. So the strong excision property applies and

the result follows.

(3.38) Lemma. Assume that the bottom row of the diagram

(3.39)

ı∗−−−→ H∗(X,Z)(ξ)∗−−−→ H∗(A,B)

δ∗η∗−−−→ H∗+1(X, Y )ı∗+1

−−−→∥∥∥∥ ξ∗

x∼= ∥∥∥∥ı∗−−−→ H∗(X,Z)

∗−−−→ H∗(Y, Z)δ∗−−−→ H∗+1(X, Y )

ı∗+1

−−−→

is exact, ξ∗ : H∗(Y, Z) → H∗(A,B) is an isomorphism and η∗ := (ξ∗)−1.Then the upper row is exact.


Proof. Let a ∈ Im ı∗. Then a ∈ Ker ∗ equivalently a ∈ Ker (ξ∗∗) since ξ∗ isan isomorphism. If b lies in Im(ξ∗∗) then b = ξ∗∗a for some a ∈ H∗(X,Z)and ∗a = η∗b which means that η∗b ∈ Ker δ∗ and b ∈ Ker δ∗η∗. Thisreasoning can be reverted. And at last if c = δ∗η∗b for some b, then c ∈ Im δ∗

so c ∈ Ker ı∗+1.

The following lemma is a consequence of well known theorem from linearalgebra. The proof can be found for instance in [41].

(3.40) Lemma. If E f−→ Fg−→ G is an exact sequence of homomorphisms

of R-modules, then

rankF = rank Im f + rank Im g.

Assuming that the modules Hq(XHA) are of finite rank we define theformal power series taking values in U(G)

PG(t,X,A) :=∑

(H)∈Φ(G)

∞∑q=0

βq(H)(X,A)tq

uG(H).

If βq(H)(X,A) = 0 for q sufficiently large and for all (H) ∈ Φ(G) then we callit Poincaré polynomial of the pair (X,A). Notice that PG( · , X,A) can beviewed as an element of the polynomial ring U(G)[t].

(3.41) Proposition. If X0 ⊂ X1 ⊂ . . . ⊂ Xm is a filtration of compactG-sets, then there exists QG(t) =

∑(H)

(∑qρq(H)t

q)uG(H) with all ρq(H) ≥ 0 such

that

(3.42)m∑j=1

PG(t,Xj, Xj−1) = PG(t,Xm, X0) + (1 + t)QG(t).

Proof. Fix (H) ∈ Φ(G). By Lemmas (3.37) and (3.38) we have a long exactsequence(3.43)

. . .δq−1(H)

ηq−1(H)−−−−−→ Hq(XjHXj−1)

ıq(H)−−→ Hq(XjHX0)

ξq(H)

q(H)−−−−−→ Hq(Xj−1HX0)→ . . . .

Here ı(H) and (H) are suitable inclusions, ξq(H) stands for the isomorphism

Hq(Xj−1HX0) ∼= Hq

X>(H)j ∪X(H)

j−1

G,X>(H)j ∪X(H)

0

G

.and ηq(H) is its inverse. Set ρq(H)(Xj, Xj−1, X0) := rank Im

(δq(H)η

q(H)

). The

exactness of (3.43) and Lemma (3.40) imply that

βq(H)(Xj−1, X0)

= ρq(H)(Xj, Xj−1, X0) + rank Im ξq(H)q(H)

= ρq(H)(Xj, Xj−1, X0) + βq(H)(Xj, X0)− rank Im ıq(H)

= ρq(H)(Xj, Xj−1, X0) + βq(H)(Xj, X0)− βq(H)(Xj, Xj−1) + ρq−1(H)(Xj, Xj−1, X0).


Consequently one has

βq(H)(Xj, Xj−1) + βq(H)(Xj−1, X0) = βq(H)(Xj, X0) + ρq(H) + ρq−1(H)

Multiplying the above equality by tq and summing over q ≥ 0 and (H) ∈Φ(G) one has

(3.44) PG(t,Xj, Xj−1) + PG(t,Xj−1, X0)

= PG(t,Xj, X0) + (1 + t)QG(t,Xj, Xj−1, X0),

where QG(t,Xj, Xj−1, X0) :=∑

(H)∈Φ(G)

(∞∑q=0

ρq(H)(Xj, Xj−1, X0)tq)uG(H). Sum-

ming (3.44) over 2 ≤ j ≤ m and setting QG(t) =m∑j=2

QG(t,Xj, Xj−1, X0)

we obtain desired result.

(3.45) Definition. Let X be an isolating neighborhood of a G-flow on Vand (M1, . . . ,Mm) be a G-invariant Morse decomposition of S = inv(X)(cf. Definition (1.7)). A G-invariant index filtration is a sequence N0 ⊂N1 ⊂ . . . ⊂ Nm of compact G-invariant subsets of V such that (Nk, Nk−1)is a G-index pair for Mk and (Nm, N0) is an index pair for S.

(3.46) Proposition. Every G-invariant Morse decomposition admits a G-invariant index filtration.

Proof. Let us forget for awhile that a Morse decomposition has a groupsymmetry. It is well known that every Morse decomposition admits anindex filtration N0 ⊂ . . . ⊂ Nm (cf. for instance [36, 37, 45]). Averaginga given filtration over group G we obtain a G-invariant index filtration fora G-invariant Morse decomposition. The compactness of Ni, 0 ≤ i ≤ msurvives since G is assumed to be compact (Corollary 1.1.3 in [3]).

Let S be an isolated invariant set of a G-equivariant flow φ. Define thePoincaré polynomial for the G-index of S as

PG(t, hG(S)) := PG(t, N, L),

where (N,L) is an arbitrary index pair for S.

(3.47) Theorem (Equivariant Morse–Conley–Zehnder equation). Let S bean isolated invariant set of a G-equivariant flow φ and let (M1, . . . ,Mm) bea Morse decomposition of S. Then there exists QG(t) =

∑(H)

(∑qρq(H)t

q)uG(H)

with all integer coefficients ρq(H) ≥ 0 such that

(3.48)m∑j=1

PG(t, hG(Mj)) = PG(t, hG(S)) + (1 + t)QG(t).

Proof. The proof is a straightforward consequence of the Propositions (3.41)and (3.46).


3.4.1 Poincaré polynomial for an isolated orbit. The aim of thissection is to give the relationship between the gradient equivariant degreeand the equiavariant Conley index using Theorem (3.47). In order to do this,we shall be interested for awhile in the case where the isolated invariant setS of a G-flow φ is an isolated zero orbit Gx of a gradient G-map f : V → V .

A couple lemmas concerning the action of compact Lie groups will beneeded. We start with the following proposition whose proof can be foundin [3] (Proposition 0.1.9 on page 4).

(3.49) Proposition. Let G be a compact group and H is a closed subgroupof G, then gHg−1 = H iff gHg−1 ⊂ H.

(3.50) Corollary. Let x ∈ V , H := Gx and Sx be a slice at x. If Q ⊂Sx ∩ V(H), then each point of Q is stationary under H.

Proof. Let y ∈ Q. By Theorem (3.11) we see that Gy ⊂ H. Since Gy andH are conjugate, there exists g ∈ G such that gHg−1 = Gy ⊂ H. TheProposition (3.49) implies that Gy = H.

(3.51) Lemma. Let x ∈ V , H := Gx and Sx be a slice at x. If Q ⊂Sx ∩ V(H), then (G×H Q)(H) = G/H ×Q and (G×H Q)>(H) = ∅.

Proof. By the definition G×H Q is a homogenous space of an action of thegroup H on G×Q defined by h(g, x) := (gh−1, hx). Since Q ⊂ Sx∩V(H) wehave h(g, x) = (gh−1, x) so the quotient space is G/H ×Q. We claim thatG/H × Q ⊂ V(H). Indeed, both Q and G/H ≈ Gx are contained in V(H),hence h(Hg, q) = (Hgh−1, hq) = (Hg, q) for h ∈ H, that is G/H × Q ⊂V (H). If there would exist K ! H such that k(Hg, q) = (Hg, q) for allk ∈ K, then Hg, q ∈ V (K) and it will be a contradiction, since G/H andQ ⊂ V(H). The result follows.

(3.52) Definition. We say that a G-invariant subset X0 of a G-set X isa strong G-deformation retract of X if there exists a G-homotopy r : X ×[0, 1]→ X such that the following properties holds true:

• r(x, 0) = x for all x ∈ X;

• r(x, t) = x for all (x, t) ∈ X0 × [0, 1];

• r(x, 1) ∈ X0.

If π : E → M is a G-vector bundle, then M is a strong G-deformationretract of E. Indeed, we identify M with a zero section of a bundle π : E →M , that is M = (x, v) ∈ E; v = 0 ∈ Ex. The homotopy is given by theformula r((x, v), t) = (x, (1− t)v).

(3.53) Proposition. Suppose that (f,Ω) is a ∇G-admissible and genericpair. Let φ denotes the flow generated by −f and Gx0 is an isolated zeroorbit of f such that Gx0 = inv(Ω). Then

(3.54) PG(t, hG(Gx0)) = tdimW−x0uG(H) ∈ U(G)


where (H) = (Gx0). If Gx0 is a principal regular zero orbit then the assump-tion about genericity can be removed. Recall that W−

x0stands for unstable

subspace of −Df(x0).

Proof. Suppose that (f,Ω) is a ∇G-admissible and generic. We shall con-struct an index pair for Gx0 via suitable choice of the index pair in the fiberof the bundle over an orbit Gx0. Let Ey = (τyGx0)⊥. By the Slice Theorem(3.11) the projection p : E → Gx0, where

E = (x, v) ∈ Gx0 × V : v ∈ Ex

is a smooth vector bundle isomorphic to π : G×H Ex0 → Gx0. Recall thatEx0 is aH-representation space. The subspace Ex0 ⊂ V splits into E+

x0⊕E−x0 ,

the stable and unstable subspaces corresponding to positive and negativespectrum of Df(x0). Since x0 is nondegenerate critical point, there existsan open H-neighborhood U of zero in Ex0 such that the flow is given inlocal H-coordinates ψ : U → Ex0 by the system of equations

x+ = A1x+ + g1(x)

x− = A2x− + g2(x), x = (x+, x−) ∈ E+

x0⊕ E−x0(3.55)

for |x| = max|x+| , |x−| ≤ 2 with g1,2 with Dg1,2 vanishes at zero, i.e.,|g1,2(x)| = o(|x|) as |x| → 0. Moreover one can chose the coordinates suchthat |g1,2(x)| and ‖Dg1,2(x)‖ can be as small as we want (cf. Appendix).The linear parts are chosen such that there exists λ > 0 for which thefollowing estimations hold

〈A1x+, x+〉 ≤ − λ |x+|2

〈A2x−, x−〉 ≥λ |x−|2 .(3.56)

If so, let B := x ∈ Ex0 ; |x| ≤ 1 and B− := x ∈ B; |x−| = 1. Then N :=ψ−1(B) and L := ψ−1(B−) is an index pair for the system on Ex0 . Finally,the index pair (X,A) for Gx0 is given by X := G×H N and A := G×H L.We shall use the assumption that (f,Ω) is generic. It implies that thenormal direction for V(H) is attracting. Set B0 := (x+, x−) ∈ B; |x+| = 0and B−0 := (x+, x−) ∈ B−; |x+| = 0 and next N0 := ψ−1(B0) and L0 :=ψ−1(B−0 ). There is a strong H-deformation retract of (N,L) onto (N0, L0),hence by the functoriality property of the twisted product the pair X0 :=G ×H N0 and A0 := G ×H L0 is a strong G-deformation retract of (X,A).The sets N0 and L0 are contained in V(H) an by the Lemma (3.51) we have

Hq(XHA) ∼= Hq(X0HA0) ∼= Hq((G/H ×N0)/G, (G/H × L0)/G)

∼= Hq(N0, L0) =

R, for q = k;0, else. ,

since the pair (N0, L0) is a homological pointed k-sphere, where k = dimW−x0

a dimension of a subspace composed by the reppeling directions, that is thenumber of negative eigenvalues of Df(x0). Hence P(t, hG(Gx0)) = tkuG(H).


Suppose now, that (f,Ω) is not a generic pair, but Gx0 is a regular zeroorbit and (H) is a principal orbit type, H := Gx0 . Then one can find anopen G-subset Ω0 ⊂ Ω0 ⊂ Ω such that Gx0 = inv(Ω0) and Ω0 ⊂ V(H). Theresult follows by using the same arguments as above.

(3.57) Corollary. Let (f,Ω) be a ∇G-admissible and generic pair, φ isa flow generated by −f and Gx0 is an isolated zero orbit of f such thatGx0 = inv(Ω). Then

(3.58) u(hG(Gx0)) = σ(Gx0)uG(H)

where (H) = (Gx0). The formula remains valid if (f,Ω) is a ∇G-admissiblepair and Gx0 is a principal regular zero orbit.

Proof. By the above Proposition

u(hG(Gx0)) = P(−1, X,A) = (−1)dimW−x0uG(H) = σ(Gx0)uG(H).

(3.59) Corollary. Let (f,Ω) be a ∇G-admissible generic pair, φ is a flowgenerated by −f and Gx0 is an isolated zero orbit of f such that Gx0 =inv(Ω). Then u(hG(Gx0)) = deg∇G(f,Ω). The formula remains valid if(f,Ω) is a ∇G-admissible pair and Gx0 is a principal regular zero orbit.

The G-index of Conley is additive in the following sense.

(3.60) Proposition. If S is an isolated invariant G-set, and S is a disjointunion S1 ∪ S2 of isolated invariant G-sets, then

PG(t, hG(S)) = PG(t, hG(S1)) + PG(t, hG(S2)).

Proof. Let (X,A) (resp. (Y,B)) be a G-index pair for S1 (resp. S2). SinceS1 and S2 are isolated one can chose those pair to be disjoint. It is clearthat (X ∪ Y,A∪B) is a G-index pair for S. Since the pairs in question aredisjoint and G-invariant one has

Hq((X ∪ Y )H(A ∪B)) =

Hq(X(H)/G ∪ Y (H)/G, (X>(H) ∪ A(H))/G ∪ (Y >(H) ∪B(H))/G).

Therefore, by the fact that pairs (XHA) and (YHB) are disjoint, we concludethat

Hq((X ∪ Y )H(A ∪B)) ∼= Hq(XHA)⊕Hq(YHB).

The above isomorphism implies that

rankHq((X ∪ Y )H(A ∪B)) = rankHq(XHA) + rankHq(YHB).

and the result follows (according to the fact that addition in U(G) is bycoordinates cf. Proposition (3.16)).


Gęba defined the gradient equivariant degree of a ∇G-admissible pair(f,Ω) as a class in U(G) representing the homotopy type of the G-Conleyindex hG(Ω, φf ), where φf stands for the flow generated by −f (cf. [14]).The following theorem states that one can justify this definition by compar-ison of the degree (Definition (3.26)) and the Conley index, and by usingthe equivariant Morse–Conley–Zehnder equation.

(3.61) Theorem. Let (f,Ω) be a ∇G-admissible pair and let Ω be an iso-lating G-invariant neighborhood of a flow φf generated by the equationx = −f(x); S := inv(Ω). Then

u(hG(S)) = deg∇G(f,Ω).

Proof. Firstly, we will show that S can be continued to an isolated invariantG-set of the flow given by the generic function. By the compactness of ∂Ωone can choose T > 0 such that for any x ∈ ∂Ω there is t ∈ [−T, T ] andφ(t, x) 6∈ Ω. Define Ω1 := Ω \ φ(∂Ω × [−T, T ]). It is clear that (f,Ω1) is∇G-admissible. By Lemma (3.25) there is a gradient G-map f1 : V → Vsatisfying f1(x) = f(x) for all x ∈ V \ Ω1 and the pair (f1,Ω1) is generic.Define the homotopy h : V × [0, 1] → V by the formula h(x, λ) := (1 −λ)f(x)+λf1(x) and let φλ stands for the flow generated by −h( · , λ). Noticethat h(x, λ) = f(x) for all x ∈ V \Ω1 that is φλ = φ on the set ∂Ω× [−T, T ].Therefore Ω is an isolating neighborhood for the flow φλ for λ ∈ [0, 1].The continuation property of the equivariant Conley index applies and onehas hG(S1) = hG(S), where S1 := invφf1 (Ω). Since the pair (f1,Ω) isgeneric the set S1 is composed of regular zero orbits Gx1, . . . , Gxm of thefunction f1 and flow lines between them. Moreover, the collection of orbitsM = (Gx1, . . . , Gxm) forms a Morse decomposition of S1. We choose anordering of M given by the potential ϕ1 : Ω → R, f1 = ∇ϕ1, i.e., one canorder the critical orbits in such a manner that ϕ1(Gxi) < ϕ1(Gxj) wheneveri > j. By the equivariant Morse–Conley–Zehnder equation

u(hG(S1)) = PG(−1, hG(S1)) =m∑k=1

PG(−1, hG(Gxk, φf1))

For 1 ≤ k ≤ m take open G-subsets Ωk ⊂ Ω, such that Ωi ∩ Ωj = ∅ andΩk is an isolating neighborhood for an isolated critical zero orbit Gxk. ByCorollary (3.57) one has PG(−1, hG(Gxk, φf1)) = deg∇G(f1,Ωk). Hence

u(hG(S)) = u(hG(S1)) =m∑k=1

deg∇G(f1,Ωk) = deg∇G(f1,Ω) = deg∇G(f,Ω)

by the additivity property and the homotopy invariance of the gradientequivariant degree (cf. [40], Theorem 3.2.).

(3.62) Example. As an easy example we will show, following Rybicki (cf.[44], Lemma 4.1.), how to compute the gradient equivariant degree of the


pair (−id, B), where −id : V → V , V is an orthogonal finite dimensionalrepresentation of G := SO(2) = γθ; 0 ≤ θ < 2π (γθ is given by equation(3.29)) and B stands for the unit ball in V . Let us introduce the followingirreducible representation of G. The notation is borrowed from [44]. Form ∈ N let R[1,m] := (R2, ρm), where ρm : G→ O(2) is given by

ρm(γθ)(x, y) = (x cosmθ − y sinmθ, x sinmθ + y cosmθ).

For k ∈ N set R[k,m] =⊕ki=1 R[1,m]. Similarly we define R[k, 0] =⊕k

i=1 R[1, 0], where R[1, 0] stands for trivial representation on the real line.Each orthogonal finite dimensional representation of G can be represented,up to equivalence, as V =

⊕pi=0 R[ki,mi], where ki,mi ∈ N for 1 ≤ i ≤ p,

k0 ∈ N ∪ 0 and 0 = m0 < m1 < . . . < mp.The multiplicative structure of U(G) is well known and can be expressed

explicitly (cf. [44]). For the convenience we denote the trivial subgroup ofG as Z1. If a = a0u

GG +

∑∞j=1 aju

GZj

and b = b0uGG +

∑∞j=1 bju

GZj

then

(3.63) ab = a0b0uGG +

∞∑j=1

(a0bj + ajb0)uGZj

Notice that B is an isolating G-neighborhood for the flow defined by theidentity vector field and the sphere S := ∂B is an exit set. Hence, one hasto compute u(SV ) = u(B/S) (comp. Section 4.4.1). According to Lemma(4.4) and formula (3.63) one has

u(SV ) = u(S⊕pi=0R[ki,mi]) =

p∏i=0

u(SR[ki,mi]) =p∏i=0

u(SR[1,mi])ki

Since SR[1,mi] is composed of, for instance, one 0-cell of orbit type G andone 1-cell of orbit type Zmi

the equality u(SR[1,mi]) = uGG− uGZmiholds. Also

u(SR[1,0]) = −uGG, therefore

u(SV ) = (−1)k0uGG

p∏i=1

(uGG − uGZmi)ki = (−1)k0uGG

p∏i=1

(uGG − kiuGZmi)

= (−1)k0(uGG +p∑i=1

kiuGZmi

)

By Theorem (3.61) we obtain deg∇G(−id, B) = (−1)k0(uGG +∑pi=1 kiu

GZmi

).

3.4.2 Some multiplicity results. As an application of the equivariantMCZ equation we shall prove a simple multiplicity result in the criticalpoint problem. Before we proceed to the statement of the result we brieflydescribe some special action of the cyclic group.

Let p be a prime number and k1, . . . , kn an integers relatively prime top. Consider an action of Zp on R2n ∼= Cn generated by the rotation

(3.64) ρ(z1, . . . , zn) = (e2πik1/pz1, . . . , e2πikn/pzn).


This action is free. Any nonzero z ∈ Cn has a nonzero coordinate zj andthen e2πiskj/pzj 6= zj for 0 < s < p since kj is relatively prime to p. The groupacts via isomerties hence the sphere S2n−1 is a Zp-invariant set. The orbitspace S2n−1/Zp is called the Lens space, denoted by Lp = Lp(k1, . . . , kn).In particular for p = 2 we have L = RP 2n−1. The above construction canbe performed for an arbitrary integer p > 1. We chose p prime to have astructure of a field in the set Zp, as a set of coefficients for a cohomologytheory. The cohomology groups of L with Zp coefficients are known andthey are (cf. [19], Example 3.41 on page 251)

Hq(Lp;Zp) ∼=

0, for q = 0;Zp, for 1 ≤ q ≤ 2n− 1.

The reduced cohomology denotes to be the cohomology relative to a base-point.

(3.65) Definition. Let f : V → R be a smooth G-invariant function.

• The orbit Gx is called critical orbit of f , if ∇f(x) = 0 (and conse-quently, for each y ∈ Gx, ∇f(y) = 0).

• The critical orbit Gx of f is said to be hyperbolic, if Gx is a regularzero orbit of ∇f (cf. Definition (3.22)).

(3.66) Proposition. Assume that V = R2n is a Zp-representation with theaction given by (3.64) and f : V → R is a smooth Zp-invariant function.Suppose that

(1) there exists a Zp-isolating neighborhood X0 such that 0 ∈ S0 := inv(X0)

and PZp(t, hZp(S0)) = uZp

Zp;

(2) f(x) = −12|x|2 +ϕ∞(x), in a neighborhood of the infinity and ∇ϕ∞ is

bounded.

If f has only a finite number of critical orbits, say Zpx0, . . . ,Zpxm, andall of them are hyperbolic, then there are at least 2n of them. Moreover(Zp)xk = E (E stands for trivial subgroup) for 1 ≤ k ≤ 2n (2np criticalpoints), and each number in the set 1, . . . , 2n is the Morse index of somecritical point.

Proof. Consider the negative gradient Zp-flow φf of x = −f(x). Let Dρ(V )(resp. Sρ(V )) stands for the disk (sphere) in V of radius ρ > 0. It followsfrom (2), that DR(V ) is a Zp-isolating neighborhood for sufficiently largeR, and (X,A) = (DR(V ), SR(V )) is a Zp-index pair for this flow. Noticethat XE/Zp ≈ (Lp × [0, 1])/(Lp × 1) =: CLp is a cone over Lp and(XZp ∪ AE)/Zp ≈ Lp × 0∪[Lp × 1] =: CLp is a disjoint union of thebottom and the top of the cone. One has

Hq(CLp, CLp;Zp) ∼= Hq(S(Lp) ∨ S1, pt;Zp) ∼=

0, for q = 0;Zp, for 1 ≤ q ≤ 2n.


It is easily seen, thatHq(XZp/Zp, (X>Zp∪AZp)/Zp) ∼= Hq(S0, pt) ∼= Zp forq = 0 and is zero otherwise. Thus the Poincaré polynomial of the Zp-Conleyindex of S := inv(X) is

PZp(t, hZp(S)) = uZp

Zp+ (t2n + t2n−1 + . . .+ t)u

Zp

E .

Since all equilibria are hyperbolic they form together with S0 a Morse de-composition (S0,M1, . . . ,Mm) of S. All nonzero orbits are principal, henceby Proposition (3.53), the Poincaré polynomial of hZp(Mi) is tqu

Zp

E providedthat q is the Morse index of Mi. Denote by ck the number of critical orbitsof index k. By the MCZ equation (3.48) there are nonnegative integersa0, a1, . . . such that

2n∑k=0

cktk =

2n∑k=1

tk + a0 +2n∑k=1

(ak−1 + ak)tk.

That is

c0 +2n∑k=1

cktk = a0 +

2n∑k=1

(ak−1 + ak + 1)tk.

Since a0 might be zero we have no information about c0, but ck ≥ 1 fork = 1, . . . , 2n.

(3.67) Remark. The assumption (1) of the above Proposition can beachieved by the following: f(x) = 1

2|x|2 + ϕ0(x), in a neighborhood of

zero and |∇ϕ0(x)| = o(|x|) as x → 0. Indeed, such condition implies thatthe origin is a critical point of f , and S0 = 0 is an isolated invariant set.The Zp-index pair for S0 is given by (Dr(V ), ∅), where r is sufficiently small.The pair (DE

r /Zp, DZpr /Zp) is homotopy equivalent to the pointed one point

space andHq(DZpr /Zp, ∅) ∼= Zp only for q = 0. Hence PZp(t, hZp(S0)) = u

Zp

Zp.

In the next Proposition, let G := SO(2).

(3.68) Proposition. Let V ∼= R[n + 1, 1] be a G-representation. Assumethat f : V → R is a smooth G-invariant function and

(1) there exists a G-isolating neighborhood X0 such that 0 ∈ S0 := inv(X0)and PG(t, hG(S0)) = uGG;

(2) f(x) = −12|x|2 + ϕ∞(x), in a neighborhood of infinity and ∇ϕ∞ is

bounded.

If f has only a finite number of critical orbits, say Gx0, . . . , Gxm, and allof them are hyperbolic, then there is at least n+1 of them. Moreover Gxk =E for 1 ≤ k ≤ n + 1, and each number in the set 2k − 1; 1 ≤ k ≤ n+ 1is a Morse index of some critical orbit.

Proof. As in the preceding proof we take the pair (X,A) = (DR(V ), SR(V ))as a G-index pair for the G flow of −∇f . Here we have XE/G ≈ (CP n ×


[0, 1])/(CP n × 1) =: CCP n, a cone over the complex projective spaceCP n and (XG ∪ AE)/G ≈ CP n × 0∪[CP n × 1] =: CCP n is a disjointunion of the bottom and the top of the cone. Now, we are going to use thecohomology with integer coefficient. Thus

Hq(CCP n, CCP n;Z) ∼= Hq(S(CP n) ∨ S1, pt;Z)

∼=

Z, for q ≤ 2n+ 1 odd;0, for q even.

Moreover, Hq(XG/G, (X>G ∪ AG)/G) ∼= Z for q = 0 and is zero otherwise.Therefore

PG(t, hG(X)) = uGG + (t2n+1 + t2n−1 + . . .+ t3 + t)uGE.

Applying the MCZ equation one obtains the equality

(3.69) c0 +n+1∑k=1

c2k−1t2k−1 +

n+1∑k=1

c2kt2k =

a0 +n+1∑k=1

(a2k−2 + a2k−1 + 1)t2k−1 +n+1∑k=1

(a2k−1 + a2k)t2k,

where cj is the number of critical orbits of index j and a0, a1, . . . are nonneg-ative integers. From (3.69) we read off that c0 ≥ 0, c2k ≥ 0 and c2k−1 ≥ 1for 1 ≤ k ≤ n+ 1.

We turn now to the case of the most general Z2-representation. Let Rt

(resp. Ra) be a one-dimensional Z2-representation with the trivial (resp.antipodal) action. Let V be an orthogonal representation of a group Z2

isomorphic to R`t ⊕Rk

a for k ≥ 1. Notice that a Z2-equivariant isomorphismA : V → V is of the form At ⊕ Aa, where At : R` → R` and Aa : Rk → Rk.Assume that f : V → R is an asymptotically quadratic Z2-invariant smoothfunction, i.e., there exist two symmetric linear Z2-maps A0, A∞ : V → Vsuch that

(1f ) f(x) = −12〈A0x, x〉+ ϕ0(x) and ∇ϕ0(x) = o(|x|), as x→ 0;

(2f ) f(x) = −12〈A∞x, x〉+ ϕ∞(x) and ∇ϕ∞(x) = o(|x|), as x→∞.

Clearly, if f is asymptotically quadratic, then the map ∇f is asymptoticallylinear. Moreover, assume that

(3f ) f is nonresonance at zero and infinity, i.e., both maps A0 and A∞ areisomorphisms, and

(4f ) f has only a finite number of critical Z2-orbits, x1, gx1, . . . , xn, gxn,and all of them are hyperbolic.


Consider a Z2-flow φf generated by −∇f . It follows, from the assumptionsabove, that the origin is an isolated invariant set for φf and there is an-other, maximal isolated invariant Z2-set T such that 0 ∈ T . There is adecomposition V = V +

0 ⊕ V −0 (resp. V = V +∞ ⊕ V −∞) corresponding to the

positive and negative spectrum of A0 (resp. A∞). Denote by Dρ(V ) (resp.Sρ(V )) the disk (resp. sphere) in V of radius ρ. It is clear, that the pair(Dr(V ), Sr(V

+0 )) is a Z2-index pair for 0 for r sufficiently small. Similarly,

the Z2-pair (DR(V ), SR(V +∞)), for R sufficiently large, is an index pair for

T .To proceed further we will calculate the cohomology groups of the pair

(D(V )ES(V )) using Z2 coefficients, that is the groups

Hq(D(V )E/Z2, (D(V )Z2 ∪ S(V )E)/Z2;Z2), q ≥ 0.

In order to visualize the geometry we need the concept of the join of twotopological spaces. Since we are dealing with quite friendly spaces, as disksand spheres, the task is much simpler than it might be possible in general.Given two topological spaces X and Y , the join X ∗Y is the quotient spaceX ×Y × [0, 1]/ ∼, where the equivalence ∼ is given by (x1, y, 0) ∼ (x2, y, 0)for x1, x2 ∈ X and y ∈ Y and (x, y1, 1) ∼ (x, y2, 1) for all x ∈ X andy1, y2 ∈ Y . We shall list some properties of the join which will be neededlater. (i) The join of X and a 0-sphere is homeomorphic to the (unreduced)suspension of X: S0 ∗ X ' SX; (ii) Sk ∗ S` ' Sk+`+1. Since the join isassociative it follows by induction that Sk ∗X ' Sk+1X, the (k+ 1)-foldedsuspension of X. The property (ii) implies in particular that if V and Ware two finite dimensional orthogonal G-representations, and S(V ) denotesthe sphere x ∈ V ; |x| = 1 then

S(V ⊕W ) ' S(V ) ∗ S(W ).

Notice that SV := D(V )/S(V ) ' S(V ⊕ Rt) (cf. Lemma (4.2)). The diskD(V )E is (k + `)-dimensional and contains the `-disk D(V )Z2 on whichthe group acts trivially. After collapsing the sphere S(V ) in D(V ) the`-disk D(V )Z2 becomes an `-sphere contained as a meridian in a sphereSV ' S(V ⊕ Rt) = S(R`+1

t ⊕ Rka) ' S(R`+1

t ) ∗ S(Rka). The group acts on

the join S(R`+1t ) ∗ S(Rk

a) as follows: g(x, y, t) = (gx, gy, t) = (x,−y, t) forall x ∈ S(R`+1

t ), y ∈ S(Rka) and t ∈ [0, 1]. Factoring out by the action of

Z2 we obtain S` ∗RP k−1. Collapsing away the circle S` (comming from thedisk D(V )Z2) one can see that the pair (D(V )ES(V )) is equivalent, up to ahomotopy type, to the pair (S` ∗ RP k−1, S`). We will examine the groupsHq(S` ∗RP k−1, S`;Z2) using the long exact sequence of a pair. One has anexact sequence of reduced cohomology groups

. . .→ Hq−1(S`)→ Hq(S` ∗RP k−1, S`)→ Hq(RP k−1 ∗ S`)→ Hq(S`)→ . . .

for q ≥ 0. By the suspension isomorphism one obtains Hq(S`∗RP k−1) ∼= Z2

for ` + 2 ≤ q ≤ ` + k. Substituting in the above sequence q = ` + i for


i = 2, . . . , k we obtain a short exact sequence

0→ H`+i(S` ∗ RP k−1, S`)∼=−→ Z2 → 0.

If q = `+ 1, then

0→ Z2

∼=−→ H`+1(S` ∗ RP k−1, S`)→ 0.

Hence, for k ≥ 1 and ` ≥ 0 one has

(3.70) Hq(D(V )ES(V )) ∼=

Z2, q = `+ i for i = 1, 2, . . . , k;0, else.

We are also interested in the cohomology of the pair (D(V ⊕U)ES(V )),where V is as above and U is an arbitrary Z2-representation. By the fol-lowing lemma one can reduce the task to the previous situation.

(3.71) Lemma. The pair (D(V ⊕U)ES(V )) is homotopy equivalent to thepair (D(V )ES(V )).

Proof. It suffices to show that the pairs (D(V ⊕U), S(V )) and (D(V ), S(V ))are Z2-homotopy equivalent. Identify D(V ⊕U) with D(V )×D(U) via thenatural Z2-homeomorphism. Define

p : (D(V )×D(U), S(V ))→ (D(V ), S(V )) andq : (D(V ), S(V ))→ (D(V )×D(U), S(V ))

by setting p(x, y) := x and q(x) := (x, 0). Clearly both p and q are Z2-equivariant, pq = id(D(V ),S(V )) and qp is homotopic with id(D(V )×D(U),S(V ))

via Z2-homotopy h(x, y, t) := (x, ty).

Let us now go back to the computations of the indices of 0 and T .Suppose that V +

0 = R`0t ⊕ Rk0

a and V +∞ = R`∞

t ⊕ Rk∞a . The above consider-

ations show that the Poincaré polynomials of the indices of 0 and T areof the form

PZ2(t, hZ2(0 , φf )) = t`0uZ2Z2

+ (t`0+1 + . . .+ t`0+k0)uZ2E and

PZ2(t, hZ2(T, φf )) = t`∞uZ2Z2

+ (t`∞+1 + . . .+ t`∞+k∞)uZ2E .

Notice that if x ∈ V is a nondegenerate critical orbit with isotropy groupZ2 (i.e., in fact, is a critical point), then

PZ2(t, hZ2(x)) = t`xuZ2Z2

+ (t`x+1 + . . .+ t`x+kx)uZ2E ,

where the numbers `x and kx are defined via the equality V +x = R`x

t ⊕Rkxa . On the other hand, for a nondegenerate critical orbit y, gy with the

isotropy group E one has PZ2(t, hZ2(y, gy)) = tdimV +y uZ2

E (cf. Proposition(3.53)). Here V +

x (resp. V +y ) is the unstable subspace of a linear map

3.5 Continuation of equivariant maps to a gradient 51

−∇2f(x) (resp. −∇2f(y)). If (1f )–(4f ) are satisfied, then combining allthese data with the equation (3.48) one obtains the following equalities:

t`0 +n∑i=1

ait`xi = t`∞ + (1 + t)Q1(t)(3.72)

k0∑i=1

t`0+i + Z (t) =k∞∑i=1

t`∞+i + (1 + t)Q2(t)

where Z ,Q1,Q2 are some unknown polynomials with nonnegative integercoefficients. The numbers ai for 1 ≤ i ≤ n may be one or zero. Notice thatai = 1 if xi is a critical orbit of orbit type Z2. It may happen that `xi = `xjfor i 6= j.

(3.73) Proposition. Suppose that f : V → R is a smooth Z2-invariantfunction satisfying conditions (1f)–(4f). If `∞ 6= `0, then f has at least twononzero critical points x, y ∈ V (two orbits of orbit type Z2). Additionally,one has an estimations on the Morse indices: dimV +

x ≥ `∞ and dimV +y ≥

`0 − 1.

Proof. We will examine the equation (3.72). The right-hand side of (3.72)contains the exponent `∞. Therefore there exists 1 ≤ i ≤ n such that`xi = `∞. On the left-hand side of (3.72) there is the exponent `0, hencethe polynomial (1 + t)Q1 contains two nonzero terms with exponents `0

and `0 + 1 or `0 − 1 and `0. Therefore, there exists 1 ≤ j ≤ n, suchthat `xj = `0 − 1 or `xj = `0 + 1. Consequently x := xi and y := xj arecritical points of f . The inclusions R`∞

t ⊂ V +x and R`0−1

t ⊂ V +y give us the

estimations on dimension of V +x and V +

y .

3.5 Continuation of equivariant maps to a gradient.

In this section we shall give an equivariant counterpart of the Reineckcontinuation theorem (cf. [38]) which says that an isolated invariant setof the flow given by a vector field can be continued, in a sense of Conleyindex theory, to the isolated invariant set of a gradient flow. Using thistheorem Reineck was able to compute the Z2-homology index of isolatedinvariant set in question by counting the number of connecting orbits be-tween critical points of adjacent indices. Beyond the intrinsic interest ofthe continuation theorem we will prove it in order to give a simple corollarythat G-equivariant Conley index is a homotopy type of a G-complex. Infact, besides some technical details, the proof verifies that the idea given byReineck works in a G-equivariant setting.

(3.74) Theorem. Let f : V → V be a G-equivariant vector field and φt bea G-flow generated by f . Let N be an isolating G-neighborhood for φ andS = inv(N). There is a continuous family of flows φtλλ∈[0,1] such that Nstays an isolating G-neighborhood for all φtλ, λ ∈ [0, 1], φt0 = φt and φt1 is agradient flow on a G-neighborhood of S.


Notice that according to the Proposition (3.32) we have hG(inv(N), φt) =hG(inv(N), φt1). The proof is based on the following crucial result due toRobbin and Salamon [39].

(3.75) Lemma. Let N be an isolating neighborhood of a flow φt with S =inv(N). There exists a neighborhood U of N and a smooth function ϕ : U →R such that

(3.75.1) ϕ(x) = 0 for all x ∈ S;

(3.75.2) ddtϕ(φt(x))|t=0 < 0 for all x ∈ U \ S.

Let µ be a normed Haar measure on G.

(3.76) Lemma (cf. [3], Theorem 0.3.3.). Let G be a compact group andf : G × R → R, (g, t) 7→ f(g, t) a continuous function, differentiable withrespect to variable t with a derivative ∂f/∂t(g, t) which is continuous onG× R. Then F (t) :=

∫G f(g, t) dµ(g) is also differentiable and

(3.77)dF

dt(t) =

∫G

∂f

∂t(g, t) dµ(g).

Proof of Theorem (3.74). Let ϕ : U → R be a Lyaponov function from theLemma (3.75). Since we cannot apriori assume that U and ϕ are G-invariantwe can average both U and ϕ over G. Namely, we define ϕ : GU → R by

ϕ(x) :=∫Gϕ(gx) dµ(g).

One can easily check that ϕ has properties (3.75.1) and (3.75.2). Indeed,G-invariance of S implies that ϕ vanishes on S. Also for x ∈ GU \S by theLemma (3.76) we have

d

dtϕ(φt(x))|t=0 =

d

dt

(∫Gϕ(gφt(x)) dµ(g)

)|t=0

=∫G

d

dtϕ(φt(gx))|t=0 dµ(g) < 0,

because gx ∈ GU \ S. In what follows, we shall omit the hat over ϕ andassume that ϕ : U → R is G-invariant. Let N ′ be a compact G-set such thatS = inv(N ′) and N ′ ⊂ int(N). Choose a G-invariant function ρ : V → [0, 1]such that:

ρ(x) =

1, dla x ∈ N ′;0, dla x ∈ V \N .

Define homotopy h : V × [0, 1]→ V setting

h(x, λ) := ρ(x)[(1− λ)f(x)− λ∇ϕ(x)] + (1− ρ(x))f(x).

One can check at once that

h( · , 0) = f, h( · , 1) = −ρ∇ϕ+ (1− ρ)f

h( · , 1) = −∇ϕ on N ′, h( · , λ) = f on V \N


The homotopy h generates a continuous family of flows φtλ such that φt0 = φt.Notice that if x 6∈ S, then ∇ϕ(x) 6= 0 and for x ∈ U \ S

(3.78)d

dtϕ(φt(x))|t=0 = 〈∇ϕ(φ0(x)), f(φ0(x))〉 = 〈∇ϕ(x), f(x)〉 < 0.

By (3.78) the function ϕ is a Lyaponov function for the family of flows φtλfor all λ ∈ [0, 1]

d

dtϕ(φtλ(x))|t=0 = 〈∇ϕ(x), ρ(x)[(1− λ)f(x)− λ∇ϕ(x)] + (1− ρ(x))f(x)〉

= (1− λ)ρ(x)〈∇ϕ(x), f(x)〉 − λρ(x)〈∇ϕ(x),∇ϕ(x)〉+ (1− ρ(x))〈∇ϕ(x), f(x)〉

= (1− λρ(x))〈∇ϕ(x), f(x)〉 − λρ(x) ‖∇ϕ(x)‖2 < 0.

In order to see that N (as well as N ′) is an isolating neighborhood for eachφtλ, λ ∈ [0, 1], notice that all equilibria of ∇ϕ are contained in the zero levelϕ−1(0). Then, if x ∈ N \ S and ϕ(x) ≤ 0 there exists T > 0 such thatφtλ(x) 6∈ N for some t ∈ (0, T ] by the compactness of N and the fact that ϕis strictly decreasing along the flow lines of φtλ. Similarly, if ϕ(x) ≥ 0 thenφtλ(x) 6∈ N for some t ∈ [−T, 0), T sufficiently large.

Notice that, in general, one can expect that the isolated invariant setchanges it structure, even disappears via the continuation. Of course, itcannot vanish if its index is nontrivial. It is also clear that setting Ω :=int(N) and g := ∇ϕ we obtain a ∇G-admissible pair (g,Ω). Hence by thecontinuation property of the index and Proposition (3.33) we obtain thepromised statement.

(3.79) Corollary. Let φt be a flow generated by a G-equivariant vectorfield. If S is an isolated invariant set of φt then the Conley index hG(S, φt)is a homotopy type of a finite G-complex.

Chapter 4

On the invertibility in U(G)

Let V be a finite dimensional real orthogonal representation of a Liegroup G. Denote by D(V ) and S(V ) the unit disc and the unit sphererespectively. Since G acts via isometries both D(V ) and S(V ) are G-sets,hence SV defined to be D(V )/S(V ) is a G-set. The following theorem dueto Gołębiewska and Rybicki [18] asserts that the G-complex SV representsan invertible element of the Euler ring U(G).

(4.1) Theorem. The element u(SV ) is invertible in U(G).

The aim of this chapter is to give a simple proof of the above theoremif the group G is finite and abelian. Our proof avoids all technicalities usedin the proof of the general case.

The nature of this theorem seems to lie far from the subject taken upin this thesis. However, there is a link between Theorem (4.1) and thetheory of topological methods in nonlinear analysis. Namely, by equalitydeg∇G(−id) = u(SV ), the gradient degree of the minus identity map is invert-ible in U(G). This was a basis for Gołębiewska and Rybicki to define thedegree of gradient strongly indefinite and G-invariant functionals defined ona Hilbert G-representation, see [18].

4.1 Technicalities.

Let Σ =

(x, t) ∈ V ⊕ R; |x|2 + (t− 1)2 = 1. We will regard Σ as a

pointed space with distinguished point p = (0, . . . , 0, 2).

(4.2) Lemma. SV is G-homeomorphic to Σ (as a pointed G-spaces).

Proof. Define ξ : SV → V ∪ ∞ by

ξ(x) :=

|x|1−|x|x, for x ∈ D(V ) \ S(V );∞, for x ∈ S(V ).

Clearly ξ is a G-homeomorphism. The inverse is given by

ξ−1(y) :=

|y|1+|y|y, for y 6=∞;[S(V )] , for y =∞

Consider the stereographic projection π : V ∪ ∞ → Σ given by the for-mula: ∞ 7→ p and

π(x1, . . . , xn) :=

(4x1

|x|2 + 4, . . . ,

4xn

|x|2 + 4,

2 |x|2

|x|2 + 4

).

4.2 Self-invertibility of SV 55

The natural G-action on Σ inherited from the action on V ⊕R coincide withthe action defined by

g ∗ z := π(gπ−1(z)), z 6= p

g ∗ p := p.(4.3)

Now it is obvious that π is G-equivariant. The composition π ξ gives usdesired G-homeomorphism.

From now on we will identify SV with Σ via the aboveG-homeomorphism.Recall that a smash product of X and Y with base points x0 and y0 respec-tively is a quotient (X × Y )/(X ∨ Y ), where X ∨ Y = X ×y0 ∪ x0× Yis a wedge of X and Y .

(4.4) Lemma. SV⊕W is G-homeomorphic to SV ∧ SW .

Proof. Let π : V ⊕W → SV⊕W \ p be the stereographic projection. Setϕ : SV ∧ SW → SV⊕W as

ϕ([x, y]) :=

π(x, y), for (x, y) 6∈

[SV ∨ SW

];

p, else.

and define ψ : SV⊕W → SV ∧ SW to be

ψ(z) :=

[π−1(z)], for z 6= p;[SV ∨ SW

], for z = p.

It is easy to see that both ϕ and ψ are G-equivariant maps. Moreover

(ψ ϕ)([x, y]) = ψ(π(x, y)) = [π−1(π(x, y))] = [x, y]

if [x, y] 6=[SV ∨ SW

]and

(ϕ ψ)(z) = ϕ([π−1(z)]) = π(π−1(z)) = z,

provided z 6= p. The same equalities holds true for [x, y] =[SV ∨ SW

]and

z = p. This completes the proof.

4.2 Self-invertibility of SV .

Notice that each real finite dimensional orthogonal G-representation (Gfinite and abelian) is of the form V = V1⊕. . .⊕Vk, where Vi’s are irreducibleeither one or two dimensional. If dimV = 1 then the action is determinedby a homomorphism G→ Z2. In the case dimV = 2 the action is given bya homomorphism G→ SO(2).

(4.5) Lemma. Let V = (R2, ρ), where ρ : G → O(2) be an irreduciblerepresentation of a compact Lie group G. Then either V = VG or thereexists a closed subgroup H ⊂ G,H 6= G such that V \ 0 = (V \ 0)(H).


Proof. Let H := Gx. It is clear, that for all x ∈ V one has Ker ρ ⊂ H. Letx ∈ V \ 0 and suppose that g ∈ H and ρ(g) 6= id. Each element of thegroup O(2) has the form

Rθ =

[cos θ − sin θsin θ cos θ

]or Tθ =

[− cos θ − sin θ− sin θ cos θ

]for θ ∈ [0, 2π). Hence, it has to be θ = 0. It contradicts the fact thatρ(g) 6= id. This implies Ker ρ = H, and each point beside the origin has anorbit type (H), except the case when G acts trivially.

(4.6) Lemma. Suppose that V is a finite dimensional orthogonal represen-tation of a finite abelian group G. Then u(SV⊕V ) is the identity in U(G).

Proof. We will distinguish two cases. Firstly, suppose that dimV = 1.Let 0 6= x ∈ V and H := Gx. Notice that the index of H in G equals2. Then V ⊕ V has the following cell decomposition: ν(G, 0) = 1 andν(H, 1) = ν(H, 2) = 2. Therefore, by Proposition (3.18) u(SV⊕V ) = uG(G).

If dimV = 2, then according to Lemma (4.4) u(SV⊕V ) = u(SV )2. If0 6= x ∈ V and H := Gx, then due to Lemma (4.5) the G-cell decompositionof V satisfies: ν(G, 0) = 1 and ν(H, 1) = ν(H, 2). It implies that u(SV ) =uG(G).

(4.7) Theorem. Suppose that V is a real finite dimensional orthogonalrepresentation of finite abelian group G. Then u(SV ) is self-invertible inU(G), i.e.,

u(SV )−1 = u(SV ).

Proof. It is an immediate consequence of Lemma (4.6).

(4.8) Example. Let G = Z2 ⊕ Z2 = e, g1, g2, g3. There are three sub-groups of G isomorphic to Z2:

Z(1)2 = e, g1, Z(2)

2 = e, g2, Z(3)2 = e, g3.

Since each element e 6= g ∈ G has rank 2, therefore all irreducible repre-sentations of G are one-dimensional. Denote them by (Vi, ρi) (see table ofcharacters). Let Φ(G) =

G,Z(1)

2 ,Z(2)2 ,Z(3)

2 , E. It is easily seen that

u(SV0) = −uGG, u(SVi) = uGG − uGZ(i)2

for i = 1, 2, 3.

Observe that u(SV1⊕V1) = u(SV1)2 = uGG − 2uGZ(1)2

+ (uGZ(1)2

)2. On the other

hand, by a direct computation one has u(SV1⊕V1) = uGG. Thus (uGZ(1)2

)2 =

2uGZ(1)2

, and u(SV1)2 = uGG, i.e., u(SV1) is a self-invertible element of U(G).

The same arguments holds for the remaining spheres. Let V =4⊕i=1

V mii be

an arbitrary representation of a group G. Then u(SV ) =∏4i=1 u(SVi)mi is

invertible, since u(SVi)mi is uGG or uGG− uGZ(i)2

depending on mi is even or oddrespectively.


e g1 g2 g3

χ1 1 1 1 1χ2 1 -1 1 1χ3 1 1 -1 1χ4 1 1 1 -1

Table 4.1: Characters of irreducible representations of G = Z2 ⊕ Z2

The following example attest to Theorem (4.7) does not hold for infinitegroups.

(4.9) Example. Let G = SO(2) and let V = R[1, 1] be an orthogonaltwo-dimensional representation of G (cf. Example (3.62)). Here Φ(G) =Zk; k = 0, 1, . . ., where Z0 := G, and Z1 := E. The sphere SV , a one-pointcompactification of V has the following G-cell decomposition: ν(0,Z0) = 1and ν(1,Z1) = 1. Therefore u(SV ) = uGZ0

−uGZ1. Using the following relations

uGZ0uGZi

= uGZiand uGZi

uGZj= 0 for i 6= j (cf. (3.63)) one can obtain that

u(SV )−1 = uGZ0+ uGZ1

.We will show explicitly, that an element uGZ0

+ uGZ1cannot be repre-

sented by the G-homotopy type of the one-point compactification of any G-representation. Notice that if R is a one-dimensional trivialG-representation,then u(SR) = u(S1) = −uGZ0

and

u(SV⊕R) = u(SV ∧ S1) = u(SV )u(S1) = −uGZ0+ uGZ1

.

We easily find out that uGZ0= u(S2), where S2 is a compactification of a two

dimensional trivial G-representation. According to the formula u(X ∨Y ) =u(X) + u(Y ) one has

uGZ0+ uGZ1

= u(SV⊕R ∨ S2 ∨ S2),

i.e., u(SV )−1 = u(SV⊕R ∨ S2 ∨ S2).

Chapter 5

Appendix

Let V be an orthogonal finite dimensional representation of a compactLie group G. Assume that Φ: V → R is a smooth G-invariant functionand the origin is a nondegenerate critical point of Φ. It is rather standardfact, that near the origin the G-flow given by an equation x = −∇Φ(x) isequivalent to the G-flow given by

x+ = A1x+ + g1(x)

x− = A2x− + g2(x), x = (x+, x−) ∈ V + ⊕ V −(5.1)

where |g1,2(x)| = o(|x|), the norms |g1,2(x)| < τ and ‖Dg1,2(x)‖ < τ , whereτ is arbitrary small. The linear maps A1,2 : V → V are such that

〈A1x+, x+〉 ≤ − λ |x+|2

〈A2x−, x−〉 ≥λ |x−|2 .(5.2)

for some λ > 0. Here V + (resp. V −) denotes the eigenspace of the Hessian∇2Φ(0) corresponding to the positive (resp. negative) eigenvalues.

For the sake of completeness we include the proof and next we will showhow to find the G-index pair for an isolated zero.

The equivalence above means that there is a G-neighborhood U 3 0 anda G-homeomorphism h : U → h(U) such that h(0) = 0 and h maps orbits inU of the first system onto orbits of the second one preserving the directionin time. In particular, such an equivalence takes place when the secondsystem is obtained by the smooth (diffeomorphic) G-equivariant change ofcoordinates y = h(x), i.e., the flows defined by x = f(x) and y = g(y) areequivalent provided that f(x) = (Dh(x))−1g(h(x)).

Let A := ∇2Φ(0). Then ∇Φ(x) = Ax + φ(x) where |φ(x)| = o(|x|) as|x| → 0. Choose a Jordan basis vi of V such that A with respect to vihas a matrix representation

A =

(A1 00 A2

),

where A1 := A|V + and A2 := A|V − are a diagonal matrices. The inequalities(5.2) are clear since A1 (resp. A2) has only negative (resp. positive) entrieson the main diagonal. For an element x = (x+, x−) ∈ V + ⊕ V − define itsnorm |x| = max |x+| , |x−|. The linear change of coordinates x 7→ εx,gives us an equivariant map Fε(x) = Ax+ φε(x), where φε(x) := 1

εφ(εx).

59

(5.3) Lemma. For any τ > 0 there exists an ε > 0 such that |φε(x)| < τand ‖Dφε(x)‖ < τ uniformly for x ∈ B2(0).

Proof. Fix τ > 0. Since |φ(x)| = o(‖x‖) as x tends to 0 there is a δ > 0 suchthat |φ(εx)|

|εx| ≤τ2provided that |εx| < δ. Let ε be chosen such that |εx| < δ.

Then|φε(x)| = 1

ε|φ(εx)| = |x| |φ(εx)|

|εx|≤ 2

τ

2= τ

The derivative of φε(x) is Dφε(x) = Dφ(εx). Since Dφ(x) is continuousand Dφ(0) = 0 for any τ > 0 one can take δ1 > 0 such that ‖Dφ(εx)‖ ≤ τif only |εx| < δ1. Taking ε small enough we are done.

In order to find the index pair for the isolated invariant set 0 for theflow given by (5.1) we proceed as follows. Let N be the square |x| ≤ 1.It is easily seen that N is G-set since the action is orthogonal. If |x+| ≥|x−| then d/dt |x+|2 = 2〈x+, x+〉 = 2〈A1x+, x+〉 + 2〈g1(x), x+〉 ≤ −λ |x+|2provided that 2τ ≤ λ. The same argument shows that if |x+| ≤ |x−|then d/dt |x−|2 ≥ λ |x−|2. Therefore, the flow of (5.1) leaves the squareN via the set N− = x ∈ N ; |x−| = 1 while the entrance set is N+ =x ∈ N ; |x+| = 1. That is the pair (N,N−) is a G-index pair for 0.To see that N− is a G-set suppose, to the contrary, that x ∈ N− andgx ∈ |x| = 1 \ N− for some g ∈ G (the sphere |x| = 1 is obviously aG-set). If so, there exists sufficiently small t > 0 such that φ(0,t)(gx) ⊂ Nwhile φ(0,t)(x) 6⊂ N and by the G-invariance of N one has gφ(0,t)(x) 6⊂ N .But this contradicts that φt is a G-map.

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topological invariants for equivariant flows: conley index

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