DIVISOR SPACES ON PUNCTURED RIEMANN SURFACES

In this paper, we study the topology of spaces of n-tuples of positive divisors on (punctured) Riemann surfaces which have no points in comm on (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at th e E-2 term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163-221. We also study the homotopy type of cer tain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were stud ied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, w ho conjectured that they split. In this paper, we show that the splitt ing does occur provided we invert the prime two.