Title

Rational Functions, Labelled Configurations, And Hilbert Schemes

Document Type

Article

Publication Date

6-1-1991

Published In

Journal Of The London Mathematical Society

Abstract

In this paper, we continue the study of the homotopy type of spaces of rational functions from S2 to CP(n) begun in [3, 4]. We prove that, for n > 1, Rat(k)(CP(n)) is homotopy equivalent to C(k)(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 C(k)(R2,S2n-1) and the Hilbert scheme moduli space for Rat(k)(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 types of Rat2(CP1) and C2(R2,S1) and show how they differ.