Rational Functions, Labelled Configurations, and Hilbert Schemes

In this paper, we continue the study of the homotopy type of spaces of rational functions from S to CP begun in [3,4]. We prove that, for n > 1, Ratfc(CP ) is homotopy equivalent to Ct(R , S""), the configuration space of distinct points in R 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(U , S"") and the Hilbert scheme moduli space for Ratt(CP ) defined by Atiyah and Hitchin [1]. When n — 1, these results no longer hold in general, and, as an illustration, we determine the homotopy types of RatjOC/*) and C0*,S) and show how they differ. Let Ratfc(CP ) denote the space of based holomorphic maps of degree k from the Riemann sphere S to the complex projective space CP. The basepoint condition we assume is that/(oo) = (1 ,1 , . . . , 1). Here we are thinking of S as C U oo, and we are describing the basepoint in CP in homogeneous coordinates. Such holomorphic maps are given by rational functions: Ratt(CP ) = {(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(CP ) 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, X for N large.) One of the goals of this paper is to 'desuspend' this result by identifying the actual homotopy type of Ratfc(CP ). We shall prove the following. Let C(U, Y) denote the space of all configurations of distinct points in U with labels in Y. That is, where F(U, q) = {(xv..., xq): xt e IR , 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) 509-528 510 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(U, Y) is homotopy equivalent to the based loop space Q,T,Y = {f:S >ZY:f(oo) = *eY}. Now let Ck(U , Y) c C(U, Y) denote the subspace of configurations of length at most k. That is, Ck(U ,Y)=\jF(n,q)x1J°/~. THEOREM 1. For n > 1, there is a natural homotopy equivalence hk: Ck(U , S"") ^ Rat^CP") which extends to a homotopy equivalence h: C(U,5"") cs Q5" -> Qk]CP . That is, the following diagram homotopy commutes. In this diagram, a is induced by the May-Milgram-Segal equivalence mentioned above, Clk]CP n denotes the connected component ofQCP of degree-k maps, and the righthand vertical arrow is the natural inclusion. REMARKS. (1) The asymptotic statement, that lim Rat^CP") ~ QS, 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^CP) and C^R, S) are equivalent to S), 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^CP) and C2(U , S), describing their homotopy types and proving that they are not equivalent. Now, the geometry of the rational function spaces Ratfc(CP ) has been much studied recently, mostly in connection with spaces of SU(2) monopoles [6]. In particular, Atiyah and Hitchin [1] gave Ratfc(CP ) (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(CP ) 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