Rational Functions, Labelled Configurations, and Hilbert Schemes

Abstract

In this paper, we continue the study of the homotopy type of spaces of rational functions from S2 to CPn begun in [3,4]. We prove that, for n > 1, Ratk(CPn) is homotopy equivalent to Ck(R2, S2n−1), the configuration space of distinct points in R2 with labels in S2n−1 of length at most k. This desuspends the stable homotopy theoretic theorems of [3, 4]. We also give direct homotopy equivalences between Ck(R2, S2n−1) and the Hilbert scheme moduli space for Ratk(CPn) 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 Rat2(CP1) and C2(R2,S1) and show how they differ.

The homotopy type of spaces of resultants of bounded multiplicity

“…We also recall the stable result obtained by F.Cohen-R.Cohen-B.Mann-R.Milgram and its improvement due to R.Cohen-D.Shimamoto ( [5], [6], [9]).…”

“…[14]), and in [17] it was shown that there is a homotopy equivalence 1 (1.2) SP d n (C) ≃ Hol * ⌊ d n ⌋ (CP 1 , CP n−1 ), if n ≥ 3 (see Theorem 1.7). The argument makes use of the existence of a C 2 -operad actions on the spaces d≥0 SP d n (C) and d≥0 Hol * d (CP 1 , CP n−1 ) ( [4], [9], [17]). Recently Benson Farb and Jesse Wolfson [11] made a remarkable discovery.…”

Relationship between polynomials with multiple roots and rational functions with common roots

“…(i) The right diagram for l = 0 and n ≥ 3 is Theorem 1.4 in [7]. By a similar argument, we can prove (i).…”

“…is proved in [9] and the second is proved in [7]. The condition n ≥ 3 in (2.3) implies that each space is simply connected.…”

On the Space of Holomorphic Maps From the Riemann Sphere to the Quadric Cone

“…This result permits an explicit computation of the homology of this space. More recently, Cohen and Shimamoto (in [5]) have determined the homotopy type, when n > 1. The results of [4] have been extended to the case of Grassmannians and flag manifolds in [14], [15].…”