The Cohomology Ring of the Space of Rational Functions

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.