  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.
