The Cohomology Ring of the Space of Rational Functions

TL;DR

This paper investigates the cohomology rings of spaces of rational functions and related configuration and braid spaces, revealing differences in their algebraic structures and localization properties for various degrees.

Contribution

It demonstrates that, except for specific cases, the F_2-cohomology rings of these spaces are not isomorphic, and shows homotopy equivalences of their HF_2-localizations.

Findings

F_p-cohomology rings are isomorphic for all three spaces when p is an odd prime.

F_2-cohomology ring of Rat_k differs from that of Bbeta_{2k} or C_k for most k.

Bbeta_{2k} and Bbeta_{2k+1} have homotopy equivalent HF_2-localizations.

Abstract

Let Rat_k be the space of based holomorphic maps from S^2 to itself of degree k. Let beta_k denote the Artin's braid group on k strings and let Bbeta_k be the classifying space of beta_k. Let C_k denote the space of configurations of length less than or equal to k of distinct points in R^2 with labels in S^1. The three spaces Rat_k, Bbeta_{2k}, C_k are all stably homotopy equivalent to each other. For an odd prime p, the F_p-cohomology ring of the three spaces are isomorphic to each other. The F_2-cohomology ring of Bbeta_{2k} is isomorphic to that of C_k. We show that for all values of k except 1 and 3, the F_2-cohomology ring of Rat_k is not isomorphic to that of Bbeta_{2k} or C_k. This in particular implies that the HF_2-localization of Rat_k is not homotopy equivalent to HF_2-localization of Bbeta_{2k} or C_k. We also show that for k >= 1, Bbeta_{2k} and Bbeta_{2k+1} have homotopy equivalent HF_2-localizations.