Abstract: We consider three spaces which can be viewed as finite-dimensional approximations to the 2-fold loop space of the 2-sphere, Ω^{2}S^{2}. These are Rat_{k}(**CP**^{1}), the space of based holomorphic maps S^{2}→S^{2}; Bβ_{2k}, the classifying space of the braid group on 2k strings; and C_{k}(**R**^{2}, S^{1}), a space of configurations of k points in **R**^{2} with labels in S^{1}. Cohen, Cohen, Mann, and Milgram showed that these three spaces are all stably homotopy equivalent. We show that these spaces are in general not homotopy equivalent. In particular, for all positive integers k with k+1 not a power of 2, the mod 2 cohomology ring of Rat_{k} is not isomorphic to that of Bβ_{2k} or C_{k}. There remain intriguing questions about the relation among these three spaces.

Since 1990, a few papers have built on this preprint, including:

J. Havlicek. The cohomology of holomorphic self-maps of the Riemann sphere. Math. Z. 218 (1995), 179–190.

D. Deshpande. The cohomology ring of the space of rational functions (2009). arXiv:0907.4412

*Photo: Susie the cat in Cambridge c. 2001.*