#### 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.

#### Recommended Citation

R. L. Cohen and Don H. Shimamoto.
(1991).
"Rational Functions, Labelled Configurations, And Hilbert Schemes".
*Journal Of The London Mathematical Society*.
Volume 43,
Issue 3.
509-528.

http://works.swarthmore.edu/fac-math-stat/81