rational functions, labelled configurations, and hilbert schemes

RATIONAL FUNCTIONS, LABELLED CONFIGURATIONS,AND HILBERT SCHEMES

RALPH L. COHEN AND DON H. SHIMAMOTO

ABSTRACT

In this paper, we continue the study of the homotopy type of spaces of rational functions from S% toCPn begun in [3,4]. We prove that, for n > 1, Ratfc(CPn) is homotopy equivalent to Ct(R2, S2""1), theconfiguration space of distinct points in R2 with labels in 5 2 "" 1 of length at most k. This desuspendsthe stable homotopy theoretic theorems of [3, 4]. We also give direct homotopy equivalences betweenCk(U

2, S2""1) and the Hilbert scheme moduli space for Ratt(CPn) defined by Atiyah and Hitchin [1].When n — 1, these results no longer hold in general, and, as an illustration, we determine the homotopytypes of RatjOC/*1) and C0*,Sl) and show how they differ.

Let Ratfc(CPn) denote the space of based holomorphic maps of degree k from theRiemann sphere S2 to the complex projective space CPn. The basepoint condition weassume is that/(oo) = (1 ,1 , . . . , 1). Here we are thinking of S2 as C U oo, and we aredescribing the basepoint in CPn in homogeneous coordinates. Such holomorphicmaps are given by rational functions:

Ratt(CPn) = {(p0, ...,pn):each/?, is a monic, degree-fcpolynomial in one

complex variable and such that there are no roots common to all the p{).

The stable homotopy type of Ratfc(CPn) was described in [3,4] in terms ofconfiguration spaces and Artin's braid groups. (Recall that the 'stable homotopytype' of a finite complex X refers to the homotopy type of the N-fold suspensionI,N X for N large.) One of the goals of this paper is to 'desuspend' this result byidentifying the actual homotopy type of Ratfc(CPn). We shall prove the following.

Let C(U2, Y) denote the space of all configurations of distinct points in U2 withlabels in Y. That is,

where F(U2, q) = {(xv..., xq): xt e IR2, xt # x}} and I 8 is the symmetric group on qletters. The relation is generated by setting

(xv...,xq)xtJitlt...,/g_15*) ~ (xlt...,xg_t)xz^pv• • •, Vx),

where * e Y is a fixed basepoint.

Received 14 September 1989.

1980 Mathematics Subject Classification (1985 Revision) 55P35.

The first author was partially supported by grants from the NSF including grant DMS 8505550through MSRI and an NSF-PYI award, the second author by a Eugene M. Lang Fellowship.

J. London Math. Soc. (2) 43 (1991) 509-528

510 RALPH L. COHEN AND DON H. SHIMAMOTO

A well-known result of May, Milgram, and Segal [9,10,11] states that, when Yis a connected CW complex, C(U2, Y) is homotopy equivalent to the based loopspace

Q,2T,2Y = {f:S2 >Z2Y:f(oo) = *eY}.

Now let Ck(U2, Y) c C(U2, Y) denote the subspace of configurations of length at

most k. That is,

Ck(U2,Y)=\jF(n2,q)x1J°/~.

THEOREM 1. For n > 1, there is a natural homotopy equivalence

hk: Ck(U2, S2""1) - ^ Rat^CP")

which extends to a homotopy equivalence h: C(U2,52""1) cs Q25l2n"1 -> Q2k]CPn. That

is, the following diagram homotopy commutes.

In this diagram, a is induced by the May-Milgram-Segal equivalence mentioned above,Cl2

k]CPn denotes the connected component ofQ2CPn of degree-k maps, and the right-hand vertical arrow is the natural inclusion.

REMARKS. (1) The asymptotic statement, that lim Rat^CP") ~ Q2S2n+1, wasproved by Segal in [12]. "**

(2) As mentioned above, the stable version of this theorem (that is, the theoremobtained by suspending each of the spaces and maps in Theorem 1 a large numberof times) was proved by the first author, F. Cohen, B. Mann, and R. J. Milgram in[3, 4].

(3) Assume that n = 1. Theorem 1 is then true when k = 1 (both Rat^CP1) andC^R2, S1) are equivalent to S1), but it is no longer true in general. Later in this paper,we focus especially on the case when k = 2, that is, on the spaces Rat^CP1) andC2(U

2, S1), describing their homotopy types and proving that they are not equivalent.

Now, the geometry of the rational function spaces Ratfc(CPn) has been muchstudied recently, mostly in connection with spaces of SU(2) monopoles [6]. Inparticular, Atiyah and Hitchin [1] gave Ratfc(CPn) (and hence an appropriate spaceof SU(«+1) monopoles of charge k, for example, hyperbolic monopoles) analgebraic-geometric description by using a fibrewise Hilbert scheme construction.This describes Ratfc(CPn) as a desingularization of the symmetric productSP*(Cx(C-{0})).

The second goal of this paper is to describe the relationship between thehomotopy theoretic description of rational functions in terms of configuration spaces

HILBERT SCHEMES 511

given in the above theorem and the geometric description in terms of Hilbert schemes.We shall describe explicit homotopy equivalences between Q((R2,52""1) and thesemoduli spaces when n > 1.

The organization of this paper is as follows. In Section 1, we prove Theorem 1 bystudying the combinatorics of the configuration spaces Ck(U

2, S2n~x) and reducing theproof to a lemma (Lemma 1.6) which states that two explicit maps

<j>k and ¥k: F(U2, k) x ^OS2""1)*"1 — * Rat,(CP»)

are homotopic. This reduction uses the notions of equalizers and mapping tori. Thesection concludes with some remarks on the case when n = 1. This theme is taken upmore fully in the following section, where we completely determine the homotopytypes of Rat2(C/>1) and C2(U

2, S1) as CW complexes in terms of cells and attachingmaps. In this regard, we might mention that the case of SU (2) monopoles of charge2 has received particular emphasis in the literature [1,7]. Finally, in Section 3, westudy the relationship between configuration spaces and Hilbert schemes.

1. The proof of Theorem 1: configurations and mapping tori

As mentioned in the introduction, the strategy behind the proof of Theorem 1 isto reduce it to the task of showing that two explicit maps are homotopic (Lemma 1.6).The idea is the following. In [3, 4] a stable homotopy equivalence

hk: ^Ck{U2, S2""1) > E00 Ratt(CPn)

was constructed by patching together certain unstable maps

gq: F(U\ 4) x i ^ 2 " ' 1 ) 9 • Rat,(CP")

using the Snaith stable splitting maps

£ 0 0 ^ = ^F{U\ q)+ A Zq{S2n~x)(9) - ^ -> Z^FiM2, q) x j ^ S 2 " " 1 ) 0 .

Here l^X denotes the suspension spectrum of X. The point was that the maps gq donot respect the basepoint relation in the definition of

ck(n\ s2"-1) = u F{n\ q) x lq(s2n-ly/ -

8-1

and hence do not directly define a map Ck(U2,52""1) -> Ratfc(CPn). Thus, the Snaith

stable splitting maps were used to construct a stable map

v gV X"DQ > V 2 2 1 ^

This, together with the fact that the composite

V £°°z)9—> V

is a stable homotopy equivalence [13], yielded a stable map

hk: Z°°Cfc((R2, S2""1) > X00 Ratfc(CPn)

which was shown to be a stable homotopy equivalence [3,4].


The first objective of this section is to show that, in order to produce an unstablehomotopy equivalence

hk: Ck(U\ S2""1) • Ratfc(CPn)

satisfying the conditions of Theorem 1, it is only necessary to show, in a sense we shallmake precise (Theorem 1.4), that the maps gq respect the basepoint relation inCk(U

2,52""1) up to homotopy. Having established this, we shall then verify that therequired homotopy conditions are in fact satisfied, provided n > 1.

Let us begin by recalling the definition of the maps

gq: F{U2, q) x z/S2"-1)' > Rat9(CP»).

The space Ratt(CPn) may be identified with the subspace of the product of symmetricproduct spaces (SPk(C))n+1 consisting of all (w+l)-tuples (f0,..., fn) 6 (SP*(C))n+1

such that there is no coordinate common to all the £t. That is, i f / = (p0,... ,/?„) is anelement of Ratfc(CPn) as in the introduction, then ^ e SPk(C) represents the roots ofpt. For instance, Rat^CP") = Cn+1-{(z,...,z):zeQ.

As described at the end of [4], the maps gq are defined in terms of some givenhomotopy equivalence v\S2n~l -> Rat^CP"); for example, the embedding

v(zli...,zn) = (z1,...,zn,0). (1.1)

(The coordinates on the left come from thinking of S2"'1 as a subspace of Cn.) Thengq:F(R2,q) x z (52""1)0 -> Ratfl(CPn) is generated out of v by applying the action ofF(U2,q) on JJJLJ Rat?(CPn). (This is the '% operad structure' of [2].)

An explicit formula for gq can be given as follows. If xe U2 = C and e > 0, let Bxt

denote the open ball about x of radius e. It will be convenient to have a family ofhomeomorphisms hx Z:C -*• Bxe depending continuously on x and e. Thus, define

0, u = 0

and, in general, hxe(y) = x + h0 e(y). Note that the embedding

(hxjn+1: Cn+1 > (BxJ

n+1 c—> Cn+1

restricts to give a mapRat^CP") • Rat^CP")

which we also denote by hxe. (It takes the roots and shrinks them inside Bxt.)Now assume that

( x ; t ) = (Xl, ...,xQ)xIff(f15..., tg)eF(U2,q) x l Q ( S 2 n ~ l y

is given. Let e = min^dle, — e^l/4}. We shall define gQ(\;t} to be a sort of'product'of the functions /jXiie(y(/<))eRatA.(CPB), but this requires some explanation.

In general, it makes sense to speak of product maps

(SP*(C))n+1 x (5P'(C))n+1

defined by componentwise ' concatenation'; that is,

..,Znrin). (1.2)

HILBERT SCHEMES 513

If £eRatfc(CPn) and neRat^CP11), it does not necessarily follow that £-rj will be inRatfc+,(CPn). However, this will follow if, say, the roots of all the ^ are disjoint fromthe roots of all the n}. In the case at hand, the roots of each hx >e(y(O) ^e within thee-ball about x( and, by construction, these balls are pairwise disjoint. Hence, if

1 x ... x (SP\C))n^ >(SPQ(C))n+1

denotes the #-fold iterated concatenation, then

gg(x; t) = rthXi,Mh)), • • •, hXqMQ)) 0 -3)

is actually an element of Rat9(CPn), and this is our definition of the map gg.

These maps may be used to prove Theorem 1 in the following way. For q ^ k, let

j : Rat9(C/>") c—> Ratfc(CPn)

be Segal's embedding [12].

THEOREM 1.4. Let /ifc:Cfc(R2,52n"1)-^Ratfc(CPn) be any map that makes the

following diagram homotopy commute.

j

CJU2, S2""1) • Ratfc(CP")

KThen hk is a homotopy equivalence satisfying the conditions of Theorem 1.

Proof. Let hk: Q(IR2, S2""1) -* Ratfc(CPn) satisfy these properties. In [3, 4], it wasshown that the stable composition

V e V g jV S°°Z)5—* V ZcoF(IR2,?)xl8(S2n-1)9 • V Z00 Rat9(CPn) * 200 Ratfc(CPn)

is a stable homotopy equivalence. Now, by the Snaith splitting [13], the composition

V ^Dq > V Z ^ R * , ? ) * £ OS2""1)* >Y.™Ck(U\S*n-x)qik g^k

is also a stable homotopy equivalence. Hence, by the commutativity of the abovediagram,

hk: Ck(U2, S2""1) > Ratfc(CPn)

induces a stable equivalence of suspension spectra and therefore in particular inducesan isomorphism in homology.

Now, in order to prove that hk is actually a homotopy equivalence, we shall, fortechnical reasons, study a 'fattened up' version of Cfc(IR

2,52""1), constructed as amapping torus. We begin by recalling this construction.

17 JLM43


Let/and g: Z-> Y be two continuous, basepoint preserving maps. Recall that theequalizer of/and g is the quotient space

E(f,g)=Y/~,

where the equivalence relation is given by setting J{x) ~ g(x)e Y for all xeX. E(f,g)is universal with respect to the property of factoring maps from Y whose compositionswith / and g are equal.

The homotopy theoretic version of the equalizer is the mapping torus

M{f,g)=Y[)XxI/~,

where / is the unit interval and the equivalence relation is given by (x,Q) ~J{x)e Yand (x, 1) ~ g(x)e Y for all xeX. Also, (x0, t) ~ yoe Y, where x0 and y0 denote thebasepoints in X and Y, respectively, and t is an arbitrary element of /.

Notice that M(f,g) has the property that any map h: Y->Z satisfying hofcahogcan be extended to a map H:M(f,g)->Z. Choices of homotopies correspond tochoices of extensions.

Consider the obvious projection map

p:Ai(f,g) >E(f,g)

defined to be the identity map on 7 c M(f,g) and given by p(x, t) =J{x) ~ g(x)sE{f,g) for (x,t)eXxI a M(f,g). A standard result is that, when either/or g is acofibration, this projection map is a homotopy equivalence.

We now observe, as in Lewis's thesis [8], that Ck(Un, Y) can be viewed as an

equalizer. Namely, define maps

a and fi: U F(Un, q) x V l Y^ > JJ F{U\ q) x ZQ Y'

as fo l lows . L e t (xlt ...,xq)xz (*!>-•->^-i)eF(^n,q)*i Y"'1. H e r e 1,^ ac t s o n the

first q—\ coordinates of F(Un,q). Define

a((x1 5 . . . ,xQ) x (tlt..., tq_j) = (xlt ...,xg)x (tlt..., tq_x, *)}

GF(Un,q)xl9YQ,

fi((xlt ...,XQ)X (tv ..., / ,_!» = (Xv ..., Xg_J X (/x, . . . , t^(1.5)

Here *e Y is the basepoint. In the important case when Y = 512""1, we take • equalto (1,0,...,0).

Observe that the basepoint relation in the definition of Cfc([Rn, Y) is simply the

assertion thatCk(U

n, Y) =

Now define Ck(Un, Y) to be the mapping torus, M{<x,P). If the basepoint in Y is

nondegenerate, the map a is a cofibration and therefore the projection map

p:Ck(Un,Y) >Ck(U

n,Y)

is a homotopy equivalence. (See [8].)

HILBERT SCHEMES 515

We now use this construction to complete the proof of Theorem 1.4. Since theprojection

is a homotopy equivalence and since the inclusion

2» 4)x x.(S2n~x)q * Q(IR2,512""1)

is a cofibration, the composition hkop is homotopic to a map

hk: Ck(U2, S2"-1) > Ratfc(CPn)

that makes the following diagram (strictly) commute.

UsQ M

JJ^.Rat^CP")

j

>Rat,(CPn)

Moreover, this cofibration property also allows the extension of hk to a map

h: C(U2, S2n~l) = lim Ck(U2, S2"-1) ^ lim Ratfc(CPn) =

k-*co k->oo

so that the following diagrams commute.

C (M2 C2""1^

h

•Rat^CP")

Rat-.CCP")

By the argument using the Snaith splitting given above, we see that h is ahomology equivalence between two simple spaces (they are both equivalent to theloop space Q2S2n+1). Hence h is a homotopy equivalence.

Now, by results of Segal [11,12], the inclusion maps Ck(U2, S2"-1) c* C(U2,52""1)

and Ratfc(CPn) c* RatJCP") are both A:(2«— 1)— 1 -connected. Hence, hk is ak(2n—l) — l-connected map and a homology equivalence. Thus, hk is a homotopyequivalence and satisfies Theorem 1. This completes the proof of Theorem 1.4.

In the introduction, we stated that a goal of this section was to reduce the proofof theorem 1 to showing that two specific maps are homotopic. We are now ready tomake this precise.

17-2


LEMMA 1.6. Ifn>\, then for each q^ 1 the compositions

a gq<f>g: F(U2, q) x ^ ( S 2 - 1 ) * - 1 > F{U\ q) x ^(S2""1)5 > Rat,(CPn)

and

in • CYO2 si\ \s /o2n—l\g—1 . EYICD2 n 1\ w ( Qin—\\Q—1 *.l?of ((T^ D"\y/g.ryH ,q)X^ yo ) >z*̂ lhx ,q—IJXj- ^o ) • l\.a.lg_i[fL>r )

. R a t 9 ( C nare homotopic.

We shall prove this lemma in a moment, but let us first show how Theorem 1 isan immediate consequence. So assume the truth of the lemma for now.

By the definition of the space (^(IR2, S2""1) as the mapping torus of the maps a and/3 (1.5), Lemma 1.6 implies that there exists a map

hk:Ck(U2,S2n-1) >Rat,(Cn

making the following diagram commute.

Us,

j

^ ( I R , ^ ) =

Theorem 1 now follows from Theorem 1.4 after recalling that the projection map

p: Ck(U2, S2""1) > Ck(U

2,52"-1)

is a homotopy equivalence.

Proof of Lemma 1.6. We shall prove this lemma in several steps. First, considerthe inclusion

v: F(U\ k- 1) x j^CS2""1)*"1 c—>F(M\ k) x ^(S2*-1)"-1

given by the formula

( A J • • •» tk-l) X C^l' • • • > ^ - 1 / ^ vA' • • •» '*-l> 'fc/ X v*l> • • • J Xk-l)>

where tk = (Ydt-i I'*D + 1 ^ ^2- The following two observations are immediate from thedefinitions of the maps involved.

PROPOSITION 1.7. (1) The compositions

</>k o v and y/k o v: F(U2, k -1) x ^(S2""1)*"1 c—> F(IR2, it) x ^^(S2"-1)*-1

are homotopic.(2) 77ie composition

0ov: F(U\k-1) x ^ ( S 2 - 1 ) * - 1 c—• F(IR2, A:) x ^

w homotopic to the identity.

HILBERT SCHEMES 517

Now recall that the map y/k factors through the projection map fi. Thus to proveLemma 1.6 (that is, that <j)k is homotopic to y/k) this proposition implies that it issufficient to prove that the map

0fc: F(U2, k) x ^ ( S " - 1 ) * - 1 > Ratt(CP")

factors, up to homotopy, through the projection map

0: F(U2, k) x Ifci(52n-1)fc~1 > F(U2,k-1) x j ^S 2 " " 1 )*" 1 .

We shall actually prove a slightly different, yet homotopy equivalent version ofthis statement. Let F(C*,k— 1) be the configuration space of k— 1 unordered pointsin C*, and consider the IJfc_1-equivariant homotopy equivalence

F(C*,k-\)<—>F(U\k)

given by mapping (w1,...,wk_1) to (w1,...,wk_1,0). This map induces a homotopyequivalence

F(C*,k-1) x ^(S2"-1)*-^—>F(U\ k) x ^(S 2 1 - 1 )*- 1 .

With respect to this homotopy equivalence, the projection map ft given above is givenby the inclusion

u:F(C*,k- \)xlk ^" - ly- ic >F(U2, k- 1) x ^(S2""1)*"1

induced by the inclusion C* c C = R2. Thus we have reduced the proof of Lemma 1.6to the following.

PROPOSITION 1.8. The map

</>k: F(C*, k-\)x ^ ( S ' " 1 ) * - 1 > Ratfc(Cn

extends up to homotopy through the inclusion

u:F(C*,k-1) x ^(S 2 "- 1 )*- 1 c—>F(U2,k-1) x ^ ( S 2 - 1 ) * - 1 .

We first need this proposition in the case when k = 2.

LEMMA 1.9. 02: C* x s2"-1 -». Rat2(CPn) extends, up to homotopy, to a map

f-.CxS2"-1 •Rat2(CPn).

Proof. Consider the composition

C* x 52""1 > Rat2(CPn) * RatJCP"),02 j

where j is the inclusion into Segal's limit as in the proof of Theorem 1.4. As observedin that proof, y'o 02 (and in fact jo<j>k for any k) factors through a homotopyequivalence

h: C(U2, S2"-1)-2L>Ratoo(C/3n).

Thus by the definition of the homotopy equalizer C(U2, S271'1), 7002 extends, up tohomotopy, to a map C x S2""1 -• Ratoo(CPn). Now Segal proved in [12] that theinclusion

j : Rat2(CPn)«—• Ratoo(CPn)is 3(2«—l)—l-connected. But the space C*x52""1 is homotopy equivalent to acomplex of dimension 2n(S1x52" ' 1) which for n > 1 is strictly smaller than3(2/7—_1)— 1. Thus by obstruction theory, 02:C* x 5 2 " - 1 -> Rat2(CPn) extends to amap 02: C x S2""1 -• Rat2(CPn) as required.


REMARK. Notice that the statement In < 3(2n — 1)— 1 used in this proof is theonly place in the proof of Lemma 1.9 that the assumption that n > 1 is used. Indeedas we shall see this is the only place in the proof of Lemma 1.6 that this assumptionis used.

Proof of Proposition 1.8. Fix e > 0, and let C(e) = {z: \z\ > e} be the complementof the e ball around the origin. Now since the inclusion C(e) c+ C is a cofibration, thehomotopy extension property says that in Lemma 1.9, 02: C x 52""1 -> Rat2(CP") canbe chosen so that its restriction to C(fi/2) x S2""1 is equal (not just homotopic) to 02.Furthermore, without loss of generality we can assume that the restriction of^2: C x S2""1 -• Rat2(CPn) to BE x S2"'1 (Be is the e-ball) has its image in

Rat2(CPn)e = {f=(p0.. .pn) 6 Rat2(CPn)

such that the roots of all the p \ have norm at most 2e}. This is because therestriction of 02 to (BE — {0})xS2n~1 has this property and because the inclusionRat2(CPn)e c+ Rat2 is a homotopy equivalence.

Now for £ > 0 fixed as above, let

F£C*,k-l)czF(C*,k-\)

be the homotopy equivalent subset consisting of those (k— l)-tuples {(wl,...,wlc_1):each Iw, — Wj\ > 4e}. Also, define a subspace

F:(C*,k-\)eFe(C*,k-\)

to consist of those (k— l)-tuples

{(w1,...,wlc_1)eF£C*,k-\) such that \wt\ <e for some /.}

Notice that by the norm condition on Fe(C*,k— 1), there is at most one w( with\wt\ < e. Notice, furthermore, that F+(C*,k-\) c+Fe(C*,k-\) is a I^-invariantsubspace. We therefore can consider the subspace

, ( V _ / - A C l ) X y O ) C-^/'IVL, , K II Xy IO I

There is an obvious projection map

p:F:(C*,k-1) x ^(S 2 "- 1 )*- 1 >Be x S2n~\

where \wt\ < e. There is another obvious projection map

p:Ft(C*,k-1) x Ifc_i(52n-1)fc"1 >F(C(3e),k-2) x I^S2""1)^2

given by

where again, IwJ < e. Thus we may consider the composition

pxpFt(C*,k-1) x lk_i(S

iH-1)k-1 >(BE x S2"-1) x F(C(3e),k-2) x Ifc / S ^ 1 ) * " 2

> Rat2(CPn)e x

HILBERT SCHEMES 519

Now since rational functions in the image gk_2(F(C(3e), k — 2) x Zfc (S2n~1)k~2) areof the form/= (p0, ...,pn) such that all the roots of all the p\ have norm larger than2e, the image of the above composition lies in the subspace

Rat2(CPn) ex Ratfc_2(CPn) <—> Rat2(CPn) x RatA_2(CPn).Here

") x Rat,(CPn) < > Rat4(CPn) x Rat,(CPn)

is the subspace consisting of ordered pairs (f,g) with disjoint roots. That is, none ofthe roots of any of the n+ 1 polynomials constituting/can be equal to a root of oneof the polynomials constituting g.

Now consider the map

/i: Rat^CP71) ix Rat/CP") > Rati+j(CPn)defined by

K(Po> • • • >Pn)> (?0> • • • > O ) = (Po Ml ?!>••• >Pn ?„)•

Notice that each of the pi qt is a monic polynomial of degree i+j and the {n + l)-tuple(PoQo>--->PnQn) satisfies the root condition defining Rati+;(CPn). (Notice that thepairing fi cannot be extended to a pairing on all of Rat,(CPn) x Rat/CPn) as thenecessary root condition will not in general be satisfied.)

It is now simply a check of definitions to verify that the composition

(F(C(e/2),k-1) n Ft(C*,k-1)) x ̂ ( S 1 - 1 ) * - 1 >pxp

>(Be x S2""1) x F(C(3e),k-2) x I^OS2""1)*"2

> Rat2(CPn) x Ratt_2(CPn) > Ratfc(CPn)0

is equal to the restriction

fa: (F(C(e/2),k-1) n F;(C*,k-1)) x ̂ JS^r1 Ratt(CP-)-

Notice that the intersection (F(C(e/2), k-\)0 F^(C*,k-1)) consists of (k- l)-tuples(wv...,wk_x)eFe(C*,k— 1) with e/2 <\wt\ <e for some i.

The reason for factorizing <f)k in this form is because the map

p: Ft(C*,k-1) x ̂ (S*- 1 )*- 1 >Be x S2""1

clearly extends to a map

p:F;(C,k-1) x ̂ GS2"-1)*-1 • ^ x 52n"1,

where Fe(C,k— 1) c= ^(C, A:— 1) is the homotopy equivalent subspace consisting ofthose (k— l)-tuples (wx,..., w^) eF(C, k— 1) such that each |wn — wn| > 4e and whereelements of F+(C,k— 1) have the added requirement that 0 ^ \wt\ < e for some /.

This proves that the restriction of <f>k to

(F(C(e/2),k-1) n Ft(C*,k-1)) x ^OS2""1)*-1

extends to a map

fa:Ft(C,k-1) x ̂ OS""-1)*-1 >Ratfc(CP»)

defined by replacing p by p in the above composition.


We now define the map

fk: Fe(C,k-1) x ^ ( S 2 * - 1 ) * - 1 > Ratt(CP»)

as follows. Let (\v,u) = (wv...,wk_1;u1,...>uk_l)eFE(C>k-\)xlki(S2n-1)k-1. We

define

0t(w,u) = i ,., s .- .rfc ' [0(w,u) if not.

Here 0 is directly seen to be well defined and continuous and extends the restriction

Proposition 1.8 (and hence Lemma 1.6) now follows by the equi variant homotopyequivalence between Fe(C,k—\) and F(C,k—\).

REMARK. When n = 1, the preceding argument fails. The exact point of failureis in the proof of Lemma 1.9, as remarked above. Indeed Lemma 1.6 is no longer truewhen n = 1 (apart from the trivial case of q = 1). An easy way to see this is to considerthe map y: S1 -• F(U2, q) x ^ ( S 1 ) 9 " 1 defined by

(Each coordinate here is meant to be a complex number.) The composite

S1 • F(U2, q) x ^JSy-1

is a constant map. On the other hand, the map

51 >F{U\ q) x ^ ( S 1 ) ' - 1 — y

is essential, as we shall now demonstrate.

The argument involves the notion of the 'resultant' map i?: Rat^CP1) -> C*.Namely, given j{z) = (p(z), r(z)) = p(z)/r(z) e Rat8(C/)1), where <x15..., <xff are the rootsof p and /?!,...,/?, are the roots of r, then

The resultant will feature more prominently in the arguments of the next section. Forthe time being, note that it is straightforward just to check that the composite </>g o yabove is homotopic to the map S1 -* Rat^CP1) which sends

t\-

Composing this with the resultant, one can then calculate that

7 </>Q *S1 > F(M\ q-\)x ^JS1)"'1 > Rat^CP1) > C*

is a map of degree 2(q— 1). Thus, <j>goy is essential and <J)q ^ y/g.Recall that the maps gQ were determined by the % structure on ]J*.1Rat0(CPn).

In view of Theorem 1.4 and the corresponding % structure on C(U2, S1) [9], the pointof the preceding argument is that, in the case when n = 1, the spaces Rat^CP1) and

HILBERT SCHEMES 521

Ck(U2, S1) cannot be made equivalent in a way compatible with these % structures.

In the next section, we pursue this further by showing, at least when k = 2, that thesespaces are just not equivalent at all.

2. The homotopy type o/Rat^CP1)

In this section, we study the space Rat2(CP1) in some detail, proving the followingtheorem along the way.

THEOREM 2.1. Rat2(C/>1) is not homotopy equivalent to C2{U2, Sl).

To outline briefly, we open the section in Lemma 2.2 with a description of C2((R2,

S1) and follow it up, using ideas of Segal, to obtain enough of the topology ofRatsjCCP1) to prove Theorem 2.1. We then go on to completely determine thehomotopy type of Rat^CP1); this amounts to figuring out the attaching map for thetop cell.

For the remainder of the section, Rat^CP1) will be abbreviated to Rat2.We begin with C2(U

2, S1). Stably, its homotopy type is known from the Snaithsplitting [13], namely,

S°°C2(IR2, S1) ^ I ^ S 1 V D2),

where D2 = C2(IR2, S^/CÛ2, S1) is the Z2-Moore space S2 U 2 e

z. In fact, we have thefollowing.

LEMMA 2.2. C2(U2, S1) is (unstably) homotopy equivalent to S1 V (S2 U 2e3).

Proof. Consider the cofibration sequence CÛ2, S1) c> C2(U2, S1) -• D2. In [5],

F. Cohen, Davis, Goerss, and Mahowald proved that this sequence desuspends onestage in the sense that there exists a space M and a cofibration sequence of the form

rM > CÛ2, S1) c—> C2(U

2, S1).

Of course, it is then immediate that XM ^ D2 = S2 U 2e3. But CÛ2, S1) is equivalent

to S1 and [M,^1] = H\M\T) = H\D2\T) = 0. Thus, r is nullhomotopic, and thelemma follows.

To begin comparing this result with Rat2, we recall some of the observationsregarding the resultant R\ Rat2 -> C* from [12]. There, Segal noted that R induces anisomorphism of fundamental groups so that the homotopy fibre may be identifiedwith the universal cover Rat2. Moreover, C* acts on Rat2 by multiplication on all theroots and, using (1.7),

for all AeC*. This makes R into a fibre bundle with structure group

Z4 = {/leC*:/l4 = l}.

Indeed, Segal's analysis gives a homeomorphism

so that R is actually the Z4-bundle with fibre R~\\) associated to the 4-fold coverC* -> C* and classified by a generator of [C*,BZA] = Z4. In particular, the action of


nx(C*) = Z on the fibre R~\\) <z Rat2 factors through a Z4 action, and this in turnmay be identified with the action of the structure group. In other words, if /en^C*)denotes the generator, then

f:Rat2 >Rat2 (2.3)

acts by multiplication by i = y/ — 1. (There are analogous statements for the otherRatjj. as well.)

This suffices to prove Theorem 2.1, so we give the proof now though it wouldfollow equally readily from various assertions to be made later on.

Proof of Theorem 2.1. Both Rat2 and C2{U2,Sl) have fundamental groupsisomorphic to Z, so their universal covers are classified by maps into BZ. We have justseen that the resultant plays the role of this classifying map in the case of Rat2 andthat n^BZ) acts on the universal cover by an automorphism of order 4.

On the other hand, Lemma 2.2 enables us to identify the universal cover of C2(R2,

S1) explicitly. It is the space

inside U x (S2 U 2e3) consisting of the real line with_a Moore space attached at each

integer. It is clear that n^BZ) = Z acts on C2 by ' translations'; these areautomorphisms of infinite order, even up to homotopy. For instance,

and the generator of nx{BT) acts as multiplication by /. In any case, /• does not acttrivially, in contrast to the situation for Rat2, and this completes the proof.

Of course, granted now that Rat2 is different from C2(IR2, S1), the question arises:

what is it? We answer this as follows. Consider the complex SlvS2. Its secondhomotopy group is the free module on one generator over the group ring ofn1(S

1vS*) = Z. That is,

Let/: S2 -> S1 V S2 be a map representing 1 + tens(S1 V S2). The final goal of this

section is to prove the following.

THEOREM 2.4. Rat2 is homotopy equivalent to the 3-dimensional complex

REMARK. Note that this theorem implies that n2(Rat2) = Z[t,r1]/(\whereas Lemma 2.2 implies that 7r2(C2(IR

2,lS1)) = Z[/,r1]/(2) = Z2[t,r1}.

Proof of Theorem 2.4. By the results of [3, 4], Rat2 and C2(IR2, S1) have the same

stable homotopy type. Furthermore, as noted in (2.3), Z[7r1(Rat2)] = Z[t, r 1 ] acts on7T2Rat2 = 7r2(Rat2) in such a way that t* = 1. Together with Lemma 2.2, these factsimply that Rat2 must have the homotopy type of a complex of the form

O S W S ^ l V 3 , (2.5)

where g: S2 -> S1 V S"2 satisfies the following properties.

HILBERT SCHEMES 523

(1) The stable class represented by g in n^S1 v S2) = Z2 © Z is given by the degree-2 map on the second summand.

(2) The element / 4 - 1 en^S1 V S2) = Z[t, r 1 ] become trivial in Rat2 and hence liesin the ideal generated by g.

Next, consider the sequence

£: n.iS1 v S2) • nKS1 v S2) > ns2(S

2),

where the first homomorphism is stabilization and the second is induced by pinchingoff the circle. This composite coincides with the 'augmentation' e:Z[t,t~l]-*Z,that is, the ring homomorphism determined by e(t)= 1. As a result, the ideal(g) c n^S1 V S2) must be generated by a polynomial having augmentation 2 anddividing t* — 1. It is easy to see that the only candidates for such ideals are (1 + i) and(1 + t2). On the other hand, note that

n^S1 V S2) U a+t) e3) = Z[t, r1]/^ + t) = Z,while

n^S1 VS2) U (1+t.>e3) = Z[t,n/il + t2) = Z®Z.

Thus, by comparison with (2.5), in order to finish Theorem 2.4, we are reduced toproving the following.

LEMMA 2.6. 7r2(Rat2) = Z.

Proof. For this, we use the relation between rational functions and monopolesdue to Donaldson [6]. This is explained in [1, Chapters 2 and 7] by Atiyah andHitchin, and, adopting their notation, we let M°k denote the space of SU(2)monopoles in Uz having charge k and fixed centre. These are represented by certainconfigurations of connections and Higgs fields (A, 0) on the trivial SU(2) bundle overIR3 satisfying the Bogomolny self-duality equations. Donaldson proved that there isa fibration

C* > Rat^CP1) > Ml.

Moreover, it was observed in [1] that, for the case when k = 2, M 2 is homeomorphicto the space of pairs of vectors (±JC, ±y) in IR3, where || j>|| = 1 and x-y = 0. (Thenotation is meant to indicate that all four signs give the same point in M2.) Clearly,this space is homotopy equivalent to IRP2 (viewed as the subspace x = 0), so there isa homotopy fibration

Sl >Rat2 >IRP2.

Lemma 2.6 (and thus Theorem 2.4) now follows from the induced exact sequence inhomotopy groups, making use of the fact that Tr^Ratg) = Z.

3. Hilbert schemes

In this section, we describe an explicit homotopy equivalence, when n > 1,between the configuration space Ck(U

2, S2"'1) and an algebraic-geometric model forRatt(CP") due to Atiyah and Hitchin. The model is based on the notion of a fibrewiseHilbert scheme, so we begin by recalling this construction (See [1, Chapter 6] fordetails).


Letn:Y >X

be a complex fibration between complex manifolds, with dimAr= 1. The fibrewiseHilbert scheme Y[k] is the space whose points are sheaves of cyclic 0K-modules Ssatisfying the following properties.

(1) 5 has finite support and dimH°(Y;S) = k.(2) There exist local sections 0: X-* Y with <f>*{T) = S for some cyclic sheaf T on

X.

Here, as usual, GY denotes the sheaf of germs of holomorphic functions on Y. Thespace Y[lc] of cyclic sheaves satisfying only property (1) is the A>fold Hilbert schemeof Y. Requiring property (2) as well is the fibrewise construction. There is a surjectivemapping to the symmetric product space

yw >SP*(Y)

defined by associating to Se Y[k] its support, where a point .ye Supp S is repeated dimSv times. (Note that Sv«suppsdim5ry = k> s i n c e dimH°(Y;S) = k.) For k = 2, thismap is the desingularization of SP\Y) given by blowing up the diagonal A c Y2 anddividing out by the Z2 action. Furthermore, Yl*] a Yl2] is the complement of thesingular set of the map Y[2] -• SP2(X).

In [1], it was proved that if Y = C x C*, X = C, and n: Y ->X is the projection,then the fibrewise Hilbert scheme Y[k] is homeomorphic to Rat^CP1). Furthermore,the desingularization map

yj*] >SPk(CxC*)

is defined via this homeomorphism by sending a rational function p/q G Ratt(C/>1) tothe unordered &-tuple of pairs (n, £) e C x C*, where r\ is a root of q and £ = p{rj) e C*.These pairs are repeated according to the multiplicity of the root //.

In fact, Atiyah and Hitchin proved more generally that if Y{n) = C x(Cn —{0}),X = C, and n: Y(n) -> X is the projection, then the resulting fibrewise Hilbert scheme7^](«) is homeomorphic to Ratfc(CP"). This homeomorphism is most easilyunderstood when restricted to the generic subspace

Nonsing (Ratt(CPn)) = {(p0,... ,pn) e Ratk(CPn):p0 has k distinct roots}.

We shall refer to this condition by saying that Nonsing (Ratfc(CPn)) consists of thoserational functions of degree k which have k distinct 'poles'.

Notice that, given any (p0,...,pn)eNonsing(Rat^CP")), the polynomial p0 iscompletely determined by these poles. Moreover, the remaining polynomials px,...,pn are uniquely determined by their values at the poles. The only restriction on thesevalues is that they cannot all vanish at any pole, that is, if pQ(co) = 0, then p^co) # 0for some /, 1 < i < n. Thus, an element of Nonsing (Ratk(CPn)) corresponds to aconfiguration of A: distinct points in IR2 (the roots of pQ), each point having a label inCn —{0} (the values ofp1,...,pn at that root). In other words, we have the following.

LEMMA 3.1. There is a natural homeomorphism

\ k) x Ifc(Cn - {0})fc -=-> Nonsing (Ratfc(CPn)).

HILBERT SCHEMES 525

Now, the space Nonsing (Ratfc(CPn)) is directly seen to be a subspace of Ylf\n) asfollows. Let

(x; u) = (xv ...,xk)x2t(Ml,...,uk)eF(U\k) x Efc(C*-{0})* s Nonsing(Ratfc(CP«))

be given. The ideal in Oc generated by the polynomial (z—x1)...(z — xk) defines thecyclic @c sheaf T(\; u) over X = C which has support {x15..., xk} and module structureat xteSupp T(x;u) given by

that is, the module of vector space dimension one. In a small ball Bi centred atxt G Supp T(x; u) whose closure does not contain any of the other xp define a localsection 0: C -• C x (Cn - {0}) by

0(z) = (z, ut).

By patching these sections together, this data defines a unique cyclic 0Cx<cn-{o})sheaf 5(x;u) with dimi/°(Cx(Cr i-{0}); with S(x;u)) = k having support at{(JC15 ux), . . . , (xk, uk)} and projecting to T(x;u). As in [1], the correspondence(x; u) i—> S(\; u) defines an embedding

vk: F(U\ k) x Ifc(C» -{0})* s Nonsing (Rat,(CP")) ^ ^ 7^(«) (3.2)

which extends to a homeomorphism yfc: Ratfc(CPn)We next start to connect all this up with the space C^IR2,512""1) by defining maps

for each q < k. To do this, first recall from (1.3) the map

gk: F(U\ k) x lk(S*n~r > Ratfc(CP")

used in the proof of Theorem 1. By construction, its image lies in the space of non-singular rational functions, so that it factors as a composition

Agk: F(U2, k) x ^(S2""1)* * Nonsing (Ratfc(CPn))

\k)xlk(Cn-{O})k^—

The constructions of [3, 4] may be interpreted to say that the induced map

fk: F(U2, k) x ^(S2"-1)* >F(U2, k) x Sfc(Cn -{0})*

is a homotopy equivalence. This, however, is also easy to check directly; we leave thisas an exercise for the reader.

Now, to define the maps yg, we first set yk to be equal to fk composed with theembedding vk of (3.2):

yk = »k o/fc: F(U2, k) x , (S2"-1)* > F(U2, k) x z (Cn - {0})*c—> Y[k](n).

Then, when q < k, we define yg:F(U2,q) x z (52""1)9 -*• Ylk\n) to be yk precomposedwith the usual embedding j : F(U2, q) x ^(S2""1)9 c* F(U2, k) x Xk{S2n-xf.

Taken together, this gives a map from the disjoint union

• i Y[k\n).


By the homotopy equalizing property of the mapping torus Ck(U2, S2"'1), in order to

obtain an induced map

fiC^.S*-1) >Y[k\n),

one must show that the compositions

f9: F(U\ q) x ^OS2""1)'"1 > F(U\ q) x lQ{S2n~y - ^ Y[k\n)and

ytg: F(U\ q) x ^ (S 2 * - 1 ) ' " 1 > F(U2, q-\)x ^ ( 5 " - ^ ^ Y[*\n)

are homotopic. But, for n > 1, this follows from Lemma 1.6 and the fact that

i^Rat^CP) >Y[k\n)

is a homeomorphism. Using the projection homotopy equivalence

p : Ck(U2, S2""1) > Ck(U

2,52"-1),

we thus obtain the desired result, as follows.

THEOREM 3.3. For n > 1, the maps

yg:F(U2,q)xlg(S2n-y—

factor up to homotopy through a homotopy equivalence

Finally, we close with a discussion of a model ^ . = ^ ( 1 ) for the homeomorphismtype of Ratfc = Rat^CP1) built out of the configuration spaces F(U2,q)xtJiC*)Q.(There are similar models &k(ri) for the other Ratfc(CPn), too.) The construction of &*begins with the homeomorphism F(U2,k) x r (C*)* = Nonsing (Ratfc) of Lemma 3.1.What then remains is the singular part of Ratfc, that is, those rational functions ofdegree k which have multiple poles. So, for r ^ k, let Rat£ c Ratfc denote the subspaceconsisting of those rational functions with precisely r distinct poles. Then

Rat, = U Rat;,r - l

where Rat* = Nonsing (Ratfc).Let p/q € Rat^. Thus, q is a monic, degree-fc polynomial with r distinct roots, say,

{x15 ...,xr}. If «t denotes the multiplicity of the root xt, then of course £]<r-in< = k.These roots, together with their multiplicities, uniquely determine the polynomial q.We abstract this observation as follows. Let Nrk denote the set of sequences ofpositive integers (nlf ...,nr) of length r such that Xic-i "< = -̂ The symmetric group I r

acts on Nr k. This yields the following.

LEMMA 3.4. The space of degree k monic polynomials over C that have preciselyr distinct roots is homeomorphic to

F(U\r)xXrNr<k.

The points in F(U2, r) give the roots of the polynomial and the sequence in NTi k theirmultiplicities.

HILBERT SCHEMES 527

Still assuming that p/q e Rat£, the lemma says that q may be identified with anelement

(xlt..., xr) x E ( « l 5 . . . , nr) e F(U2, r) x lf Nr< k.That is,

i-X

Now define q(z) to be the degree-r polynomial given by

q(z) = (z-x1)...(z-xr).

Using the Euclidean algorithm, we can write the numerator p(z) uniquely in the form

p(z) = q(z)J{z)+p(z),

where p(z) and j\z) are monic polynomials of degrees r and k—r, respectively. Sincep/qeRat^ the values of/? at the roots of q, namely, p(x{) = p(Xi), must be non-zero.But also p is monic of degree r, so it is completely determined by these r non-zerovalues. Observe that there is no restriction on the polynomial j{z) other than that itmust be monic of degree k—r. Hence, we have proved the following.

THEOREM 3.5. Rat£ is homeomorphic to

C- x(F(U\r)xlr((C*y x Nr,k)).

Thus, there is a homeomorphism

Rat* ̂ U C*-r x (F(U\ r) x Er((C*)r x Nrk)).r - l

Let % be the complex (J?-i C*~r x (F(U2, r) x Zr((C*)r x Nrk)). The attaching mapsbetween the various levels of ^ (that is, between the various Rat£) are quitecomplicated and are determined by what happens as distinct poles of rationalfunctions converge to form multiple poles. In this context, Theorem 1 can beinterpreted as saying that the analogous complicated attachings in Ratfc(CPn) forn > 1 can be deformed to the rather simple basepoint relation attachings of the levelsof the complex Ck(U

2, S^""1). One loses geometric information about RatA.(CPn) inthe process, but Theorem 1 says that one retains the homotopy type. On the otherhand, Theorem 2.1 may be interpreted to say that these attachings cannot be sodeformed when n = 1.

Acknowledgements. The authors are grateful to M. Atiyah for suggesting thatthey try to relate the configuration space and Hilbert scheme approaches and toS. Donaldson and N. Hitchin for helpful conversations. They are also grateful to theMathematical Institute at the University of Oxford, the first author to the HebrewUniversity of Jerusalem and MSRI as well, for their hospitality while this work wasbeing carried out.

References

1. M. ATIYAH and N. HITCHIN, The geometry and dynamics of magnetic monopoles (University Press,Princeton, 1988).

2. C. P. BOYER and B. M. MANN, 'Monopoles, non-linear a models, and two-fold loop spaces', Comm.Math. Phys. 115 (1988) 571-594.

528 HILBERT SCHEMES

3. F. R. COHEN, R. L. COHEN, B. M. MANN and R. J. MILGRAM, 'The topology of rational functions anddivisors of surfaces', Ada Math, to appear.

4. F. R. COHEN, R. L. COHEN, B. M. MANN and R. J. MILGRAM, 'The homotopy type of rationalfunctions', Math. Z. to appear.

5. F. R. COHEN, D. M. DAVIS, P. G. GOERSS and M. E. MAHOWALD, 'Integral Brown-Gitler spectra',Proc. Amer. Math. Soc. 103 (1988) 1299-1304.

6. S. K. DONALDSON, 'Nahm's equations and the classification of monopoles', Comm. Math. Phys. 96(1984) 387-407.

7. J. HURTUBISE, lSU(2) monopoles of charge 2', Comm. Math. Phys. 92 (1983) 195-202.8. L. G. LEWIS, J. P. MAY and M. STEINBERGER, Equivariant stable homotopy theory, Lecture Notes in

Mathematics 1213 (Springer, Berlin, 1986).9. J. P. MAY, The geometry of iterated loop spaces, Lecture Notes in Mathematics 271 (Springer, Berlin,

1972).10. R. J. MILGRAM, 'Iterated loop spaces', Ann. of Math. 84 (1966) 386-403.11. G. B. SEGAL, 'Configuration spaces and iterated loop spaces', Invent. Math. 21 (1973) 213-221.12. G. B. SEGAL, 'The topology of spaces of rational functions', Ada Math. 143 (1979) 39-72.13. V. P. SNAITH, 'A stable decomposition of ClnSnX\ J. London Math. Soc. (2) 7 (1974) 557-583.

Department of Mathematics Department of MathematicsStanford University Swarthmore CollegeStanford SwarthmoreCalifornia 94305 Pennsylvania 19081USA USA

rational functions, labelled configurations, and hilbert schemes

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