The Cohomology Ring of the Space of Rational Functions
Abstract
Let Ratk be the space of based holomorphic maps from S2 to itself of degree k. Let betak denote the Artin's braid group on k strings and let Bbetak be the classifying space of betak. Let Ck denote the space of configurations of length less than or equal to k of distinct points in R2 with labels in S1. The three spaces Ratk, Bbeta2k, Ck are all stably homotopy equivalent to each other. For an odd prime p, the Fp-cohomology ring of the three spaces are isomorphic to each other. The F2-cohomology ring of Bbeta2k is isomorphic to that of Ck. We show that for all values of k except 1 and 3, the F2-cohomology ring of Ratk is not isomorphic to that of Bbeta2k or Ck. This in particular implies that the HF2-localization of Ratk is not homotopy equivalent to HF2-localization of Bbeta2k or Ck. We also show that for k >= 1, Bbeta2k and Bbeta2k+1 have homotopy equivalent HF2-localizations.