Divisor spaces on punctured Riemann surfaces
Abstract: In this paper, we study the topology of spaces of -tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (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 the 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 certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2125; Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec H3C 3J7, Canada
Keywords: Riemann surfaces, symmetric products, Eilenberg-Moore spectral sequence, Whitehead product, Milgram bar construction
Received by editor(s): December 7, 1995
Additional Notes: The author holds a Postdoctoral fellowship with the Centre de Recherches Mathématiques, Université de Montréal.
Article copyright: © Copyright 1998 American Mathematical Society