The Cohomology Ring of the Space of Rational Functions

Let Ratk be the space of based holomorphic maps from S 2 to itself of degree k. Let βk denote the Artin’s braid group on k strings and let Bβk be the classifying space of βk. Let Ck denote the space of configurations of length less than or equal to k of distinct points in R with labels in S. The three spaces Ratk , Bβ2k , 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 Bβ2k 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 Bβ2k or Ck . This in particular implies that the HF2-localization of Ratk is not homotopy equivalent to HF2-localization of Bβ2k or Ck. We also show that for k ≥ 1, Bβ2k and Bβ2k+1 have homotopy equivalent HF2-localizations.