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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% to CPn begun in [3,4]. We prove that, for n > 1, Rat fc (CP n ) is homotopy equivalent to C t (R 2 , S 2 "" 1 ), the configuration space of distinct points in R2 with labels in 5 2 " " 1 of length at most k. This desuspends the stable homotopy theoretic theorems of [3, 4]. We also give direct homotopy equivalences between Ck(U2, S 2 "" 1 ) and the Hilbert scheme moduli space for Rat t (CP n ) defined by Atiyah and Hitchin [1]. When n — 1, these results no longer hold in general, and, as an illustration, we determine the homotopy and show how they differ. types of RatjOC/*1) and C0*,Sl)

Let Ratfc(CPn) denote the space of based holomorphic maps of degree k from the Riemann sphere S2 to the complex projective space CPn. The basepoint condition we assume is that/(oo) = ( 1 , 1 , . . . , 1). Here we are thinking of S2 as C U oo, and we are describing the basepoint in CPn in homogeneous coordinates. Such holomorphic maps are given by rational functions: Rat t (CP n ) = {(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 of configuration spaces and Artin's braid groups. (Recall that the 'stable homotopy type' of a finite complex X refers to the homotopy type of the N-fold suspension I,N X for N large.) One of the goals of this paper is to 'desuspend' this result by identifying 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 with labels in Y. That is,

where F(U2, q) = {(xv..., xq): xt e IR2, xt # x}} and I 8 is the symmetric group on q letters. 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 8505550 through MSRI and an NSF-PYI award, the second author by a Eugene M. Lang Fellowship. J. London Math. Soc. (2) 43 (1991) 509-528

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RALPH L. COHEN AND DON H. SHIMAMOTO

A well-known result of May, Milgram, and Segal [9,10,11] states that, when Y is a connected CW complex, C(U2, Y) is homotopy equivalent to the based loop space 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, Cl2k]CPn denotes the connected component ofQ2CPn of degree-k maps, and the righthand vertical arrow is the natural inclusion. (1) The asymptotic statement, that lim Rat^CP") ~ Q2S2n+1, was proved by Segal in [12]. "** (2) As mentioned above, the stable version of this theorem (that is, the theorem obtained by suspending each of the spaces and maps in Theorem 1 a large number of 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) and C^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) and C2(U2, S1), describing their homotopy types and proving that they are not equivalent. REMARKS.

Now, the geometry of the rational function spaces Ratfc(CPn) has been much studied recently, mostly in connection with spaces of SU(2) monopoles [6]. In particular, Atiyah and Hitchin [1] gave Ratfc(CPn) (and hence an appropriate space of SU(«+1) monopoles of charge k, for example, hyperbolic monopoles) an algebraic-geometric description by using a fibrewise Hilbert scheme construction. This describes Ratfc(CPn) as a desingularization of the symmetric product SP*(Cx(C-{0})). The second goal of this paper is to describe the relationship between the homotopy 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((R 2 ,5 2 "" 1 ) and these moduli spaces when n > 1. The organization of this paper is as follows. In Section 1, we prove Theorem 1 by studying the combinatorics of the configuration spaces Ck(U2, S2n~x) and reducing the proof to a lemma (Lemma 1.6) which states that two explicit maps k and ¥k: F(U2, k) x ÔS 2 "" 1 )*" 1 — * Rat,(CP») are homotopic. This reduction uses the notions of equalizers and mapping tori. The section concludes with some remarks on the case when n = 1. This theme is taken up more fully in the following section, where we completely determine the homotopy types of Rat2(C/>1) and C2(U2, S1) as CW complexes in terms of cells and attaching maps. In this regard, we might mention that the case of SU (2) monopoles of charge 2 has received particular emphasis in the literature [1,7]. Finally, in Section 3, we study 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 is to 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 Rat t (CP n )

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 £00^

=

^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 do not 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) -> Rat fc (CP n ). Thus, the Snaith stable splitting maps were used to construct a stable map

V X"DQ

>V

2

2

1

vg

^

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].

512


The first objective of this section is to show that, in order to produce an unstable homotopy equivalence hk: Ck(U\ S2""1)

• Ratfc(CPn)

satisfying the conditions of Theorem 1, it is only necessary to show, in a sense we shall make precise (Theorem 1.4), that the maps gq respect the basepoint relation in Ck(U2,52""1) up to homotopy. Having established this, we shall then verify that the required 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/S 2 "- 1 )'

> Rat9(CP»).

n

The space Rat t (CP ) may be identified with the subspace of the product of symmetric product 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 an element of Ratfc(CPn) as in the introduction, then ^ e SPk(C) represents the roots of pt. 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 given homotopy 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.) Then gq:F(R2,q) x z (5 2 "" 1 ) 0 -> Ratfl(CPn) is generated out of v by applying the action of F(U2,q) on JJJLJ Rat ? (CP n ). (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 of homeomorphisms 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: C n+1

> (BxJn+1 c—> C n+1

restricts to give a map Rat^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)

xlQ(S2n~ly

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(/ V Z ^ R * , ? ) * £ OS2""1)*

>Y.™Ck(U\S*n-x)

g^k

is also a stable homotopy equivalence. Hence, by the commutativity of the above diagram, hk: Ck(U2, S2""1) > Ratfc(CPn) induces a stable equivalence of suspension spectra and therefore in particular induces an isomorphism in homology. Now, in order to prove that hk is actually a homotopy equivalence, we shall, for technical reasons, study a 'fattened up' version of Cfc(IR2,52""1), constructed as a mapping torus. We begin by recalling this construction. 17

JLM43

514


Let/and g: Z-> Y be two continuous, basepoint preserving maps. Recall that the equalizer 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 compositions with / 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 Y and (x, 1) ~ g(x)e Y for all xeX. Also, (x0, t) ~ yoe Y, where x0 and y0 denote the basepoints 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 hofcahog can be extended to a map H:M(f,g)->Z. Choices of homotopies correspond to choices 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)s E{f,g) for (x,t)eXxI a M(f,g). A standard result is that, when either/or g is a cofibration, 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 andfi:U F(Un, q) x V l Y^ a s f o l l o w s . L e t (xlt ...,xq)xz

> JJ F{U\ q) x ZQ Y'

(*!>-•->^-i)eF(^n,q)*i

Y"'1. H e r e 1,^ a c t s o n t h e

n

first q—\ coordinates of F(U ,q). Define a((x 1 5 ...,x Q ) x (tlt...,

tq_j) = (xlt ...,xg)x n

(tlt...,

tq_x, *)}

Q

GF(U ,q)xl9Y , 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 • equal to (1,0,...,0). Observe that the basepoint relation in the definition of Cfc([Rn, Y) is simply the assertion that Ck(Un, Y) = Now define Ck(Un, Y) to be the mapping torus, M{Ck(Un,Y)

515

HILBERT SCHEMES

We now use this construction to complete the proof of Theorem 1.4. Since the projection 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,(CP n ) Moreover, this cofibration property also allows the extension of hk to a map h: C(U2, S2n~l) = lim Ck(U2, S2"-1) k-*co

^ lim Ratfc(CPn) = k->oo

so that the following diagrams commute.

C (M2

C2""1^

•Rat^CP")

h

Rat-.CCP")

By the argument using the Snaith splitting given above, we see that h is a homology equivalence between two simple spaces (they are both equivalent to the loop space Q 2 S 2n+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* R a t J C P " ) are both A:(2«— 1)— 1 -connected. Hence, hk is a k(2n—l) — l-connected map and a homology equivalence. Thus, hk is a homotopy equivalence 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 proof of theorem 1 to showing that two specific maps are homotopic. We are now ready to make this precise. 17-2

516


LEMMA

1.6.

Ifn>\,

then for each q^ 1 the compositions

g: F(U2, q) x ^ ( S 2 - 1 ) * - 1

a

gq

> F{U\ q) x ^(S 2 "" 1 ) 5

> Rat,(CP n )

and in • CYO2 si\ \s

/o2n—l\g—1

. EYICD2 n

y/g.ryH

yo

>z*^lhx ,q—IJXj-

,q)X^

)

1\ w

( Qin—\\Q—1

*.l?of

ô

• l\.a.lg_i[fL>r

)

((T^ D"\

)

.Rat9(Cn are homotopic. We shall prove this lemma in a moment, but let us first show how Theorem 1 is an 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(U2,52"-1)

is a homotopy equivalence. Proof of Lemma 1.6. We shall prove this lemma in several steps. First, consider the inclusion v: F(U\ k- 1) x j^CS 2 "" 1 )*" 1 c—>F(M\ k) x given by the formula X

( A J • • •» tk-l)

C^l' • • • > ^ - 1 /

^ vA' • • •» '*-l> 'fc/

^(S2*-1)"-1 X

v*l> • • • J

X

k-l)>

2

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

1.7.

(1) The compositions

k o v and y/k o v: F(U2, k -1) x ^(S 2 "" 1 )*" 1 c—> F(IR2, it) x ^^(S 2 "- 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 prove Lemma 1.6 (that is, that F(U2,k-1)

x j^S2""1)*"1.

We shall actually prove a slightly different, yet homotopy equivalent version of this statement. Let F(C*,k— 1) be the configuration space of k— 1 unordered points in C*, and consider the IJfc_1-equivariant homotopy equivalence F(C*,k-\)F(U\k) given by mapping (w1,...,wk_1) equivalence F(C*,k-1)

to (w1,...,wk_1,0).

This map induces a homotopy

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

With respect to this homotopy equivalence, the projection map ft given above is given by the inclusion >F(U2, k- 1) x ^(S 2 "" 1 )*" 1

u:F(C*,k- \)xlk ^ " - l y - i c

induced by the inclusion C* c C = R2. Thus we have reduced the proof of Lemma 1.6 to the following. PROPOSITION 1.8.

The map

k: F(C*, k-\)x ^ ( S ' " 1 ) * - 1

> Rat f c (Cn

extends up to homotopy through the inclusion u:F(C*,k-1)

x ^ ( S 2 " - 1 ) * - 1 c—>F(U2,k-1)

x ^(S2-1)*-1.

We first need this proposition in the case when k = 2. LEMMA 1.9. 0 2 : C* x s 2 "- 1 -». Rat 2 (CP n ) extends, up to homotopy, to a map f-.CxS2"-1

•Rat 2 (CP n ).

Proof. Consider the composition C* x 5 2 "" 1

> Rat2(CPn)

* RatJCP"), 02 j where j is the inclusion into Segal's limit as in the proof of Theorem 1.4. As observed in that proof, y'o 0 2 (and in fact jok for any k) factors through a homotopy equivalence h: C(U2, S2"-1)-2L>Ratoo(C/3n). Thus by the definition of the homotopy equalizer C(U2, S271'1), 700 2 extends, up to homotopy, to a map C x S 2 "" 1 -• Ratoo(CPn). Now Segal proved in [12] that the inclusion j : Rat 2 (CP n )«—• Ratoo(CPn) is 3(2«—l)—l-connected. But the space C*x5 2 "" 1 is homotopy equivalent to a complex of dimension 2n(S 1 x5 2 "' 1 ) which for n > 1 is strictly smaller than 3(2/7—_1)— 1. Thus by obstruction theory, 0 2 :C* x 5 2 " - 1 -> Rat 2 (CP n ) extends to a map 0 2 : C x S 2 "" 1 -• Rat 2 (CP n ) as required.

518


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

Proof of Proposition 1.8. Fix e > 0, and let C(e) = {z: \z\ > e} be the complement of the e ball around the origin. Now since the inclusion C(e) c+ C is a cofibration, the homotopy extension property says that in Lemma 1.9, 0 2 : C x 5 2 "" 1 -> Rat2(CP") can be chosen so that its restriction to C(fi/2) x S2""1 is equal (not just homotopic) to 0 2 . Furthermore, without loss of generality we can assume that the restriction of ^ 2 : C x S 2 "" 1 -• Rat 2 (CP n ) to BE x S 2 "' 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 the restriction of 0 2 to (BE — {0})xS2n~1 has this property and because the inclusion Rat 2 (CP n ) 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\ C* and classified by a generator of [C*,BZA] = Z 4 . In particular, the action of

522


nx(C*) = Z on the fibre R~\\) Rat2 (2.3) f:Rat 2 acts by multiplication by i = y/ — 1. (There are analogous statements for the other Ratjj. as well.) This suffices to prove Theorem 2.1, so we give the proof now though it would follow equally readily from various assertions to be made later on. Proof of Theorem 2.1. Both Rat2 and C2{U2,Sl) have fundamental groups isomorphic to Z, so their universal covers are classified by maps into BZ. We have just seen that the resultant plays the role of this classifying map in the case of Rat2 and that 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, 1 S ) 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 are automorphisms of infinite order, even up to homotopy. For instance,

and the generator of nx{BT) acts as multiplication by /. In any case, /• does not act trivially, 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 second homotopy group is the free module on one generator over the group ring of n1(S1vS*) = 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

Note that this theorem implies that n2(Rat2) = Z[t,r1]/(\ whereas Lemma 2.2 implies that 7r2(C2(IR2,lS1)) = Z[/,r 1 ]/(2) = Z2[t,r1}. REMARK.

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 on 7T2Rat2 = 7r2(Rat2) in such a way that t* = 1. Together with Lemma 2.2, these facts imply that Rat2 must have the homotopy type of a complex of the form OSWS^lV3, where g: S2 -> S1 V S"2 satisfies the following properties.

(2.5)

HILBERT SCHEMES

523

(1) The stable class represented by g in n^S1 v S2) = Z 2 © Z is given by the degree2 map on the second summand. (2) The element / 4 - 1 en^S1 V S2) = Z[t, r 1 ] become trivial in Rat 2 and hence lies in the ideal generated by g. Next, consider the sequence £: n.iS1 v S2)

• nKS1 v S2)

> ns2(S2),

where the first homomorphism is stabilization and the second is induced by pinching e:Z[t,t~l]-*Z, off the circle. This composite coincides with the 'augmentation' 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 and dividing 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

VS 2 ) 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 to proving the following. LEMMA 2.6.

7r2(Rat2) = Z.

Proof. For this, we use the relation between rational functions and monopoles due to Donaldson [6]. This is explained in [1, Chapters 2 and 7] by Atiyah and Hitchin, 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 certain configurations of connections and Higgs fields (A, 0) on the trivial SU(2) bundle over IR3 satisfying the Bogomolny self-duality equations. Donaldson proved that there is a fibration C* > Rat^CP 1 ) > Ml. Moreover, it was observed in [1] that, for the case when k = 2, M 2 is homeomorphic to the space of pairs of vectors (±JC, ±y) in IR3, where || j>|| = 1 and x-y = 0. (The notation is meant to indicate that all four signs give the same point in M 2 .) Clearly, this space is homotopy equivalent to IRP2 (viewed as the subspace x = 0), so there is a homotopy fibration Sl >Rat2 >IRP2. Lemma 2.6 (and thus Theorem 2.4) now follows from the induced exact sequence in homotopy 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(U2, S2"'1) and an algebraic-geometric model for Rat t (CP") due to Atiyah and Hitchin. The model is based on the notion of a fibrewise Hilbert scheme, so we begin by recalling this construction (See [1, Chapter 6] for details).

524


Let n:Y

>X

be a complex fibration between complex manifolds, with dimA r = 1. The fibrewise Hilbert scheme Y[k] is the space whose points are sheaves of cyclic 0K-modules S satisfying the following properties. (1) 5 has finite support and dimH°(Y;S) = k. (2) There exist local sections 0: X-* Y with *{T) = S for some cyclic sheaf T on X. Here, as usual, GY denotes the sheaf of germs of holomorphic functions on Y. The space Y[lc] of cyclic sheaves satisfying only property (1) is the A>fold Hilbert scheme of Y. Requiring property (2) as well is the fibrewise construction. There is a surjective mapping 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 dim Sv times. (Note that Sv«supps dim5r y = k> s i n c e dimH°(Y;S) = k.) For k = 2, this map is the desingularization of SP\Y) given by blowing up the diagonal A c Y2 and dividing out by the Z2 action. Furthermore, Yl*] a Yl2] is the complement of the singular 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^CP 1 ). Furthermore, the desingularization map yj*]

>SPk(CxC*)

is defined via this homeomorphism by sending a rational function p/q G Rat t (C/ >1 ) to the 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(C n —{0}), X = C, and n: Y(n) -> X is the projection, then the resulting fibrewise Hilbert scheme 7^ ] («) is homeomorphic to Ratfc(CP"). This homeomorphism is most easily understood when restricted to the generic subspace Nonsing (Rat t (CP n )) = {(p0,... ,pn) e Ratk(CPn):p0 has k distinct roots}. We shall refer to this condition by saying that Nonsing (Ratfc(CPn)) consists of those rational functions of degree k which have k distinct 'poles'. Notice that, given any (p0,...,pn)eNonsing(Rat^CP")), the polynomial p0 is completely determined by these poles. Moreover, the remaining polynomials px,..., pn are uniquely determined by their values at the poles. The only restriction on these values is that they cannot all vanish at any pole, that is, if pQ(co) = 0, then p^co) # 0 for some /, 1 < i < n. Thus, an element of Nonsing (Ratk(CPn)) corresponds to a configuration of A: distinct points in IR2 (the roots of pQ), each point having a label in C n —{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 n

525 l

Now, the space Nonsing (Ratfc(CP )) is directly seen to be a subspace of Y f\n) as follows. 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 the cyclic @c sheaf T(\; u) over X = C which has support {x 15 ..., xk} and module structure at x t eSupp T(x;u) given by

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

0(z) = (z, ut). By patching these sections together, this data defines a unique cyclic 0Cx S(\; u) defines an embedding vk: F(U\ k) x Ifc(C» -{0})* s Nonsing (Rat,(CP")) ^ ^ 7 ^ ( « )

(3.2)

n

which extends to a homeomorphism yfc: Ratfc(CP ) We next start to connect all this up with the space CÎR2,512""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 nonsingular rational functions, so that it factors as a composition gk: F(U2, k) x ^(S 2 "" 1 )*

A

* 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 ^(S 2 "- 1 )*

>F(U2, k) x Sfc(Cn -{0})*

is a homotopy equivalence. This, however, is also easy to check directly; we leave this as an exercise for the reader. Now, to define the maps yg, we first set yk to be equal to fk composed with the embedding vk of (3.2): yk = »k o/fc: F(U2, k) x , (S 2 "- 1 )*

> F(U2, k) x z (C n - {0})*c—> Y[k](n).

Then, when q < k, we define yg:F(U2,q) x z (5 2 "" 1 ) 9 -*• Ylk\n) to be yk precomposed with the usual embedding j : F(U2, q) x ^(S 2 "" 1 ) 9 c* F(U2, k) x Xk{S2n-xf. Taken together, this gives a map from the disjoint union

• i Y[k\n).

526


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 ÔS 2 "" 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 >Y[k\n)

i^Rat^CP)

is a homeomorphism. Using the projection homotopy equivalence p : Ck(U2, S2""1)

> Ck(U2,52"-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 homeomorphism type of Ratfc = Rat^CP 1 ) 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 of degree k which have multiple poles. So, for r ^ k, let Rat£ c Ratfc denote the subspace consisting 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 £]