Thanks to Claudio Gonzales, who requested it, and to MSRI Librarian Linda Riewe, who found and scanned it, my 1990 MSRI preprint “The cohomology ring of the space of rational functions” is available on my webpage.

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.*