Stability of the homology of the moduli spaces of Riemann surfaces with spin structure

Abstract

Recently, due largely to its importance in fermionic string theory, there has been much interest in the moduli spaces ~/gl-e] of Riemann surfaces of genus g with spin structure of Arf invariant e e Z/2Z. Algebraic geometers have long studied these spaces in their alternate guise as moduli spaces of pairs (algebraic curve, square root of the canonical bundle). For topologists these spaces are rational classifying spaces for spin mapping class groups. However, despite the fact that ~r162 is a finite cover of the ordinary moduli space J/4g, little is known about the topology of these spaces. In this paper we begin a study of the homology of ~'g[e] by proving that its homology groups are independent of g and e when g is adequately large (Theorem3.1). In a second paper [H4] we will compute nl(v//g[e]) and H2(./r thereby calculating the Picard group of Jlg[e]. Putting this all together we know approximately the same amount about the homology of Jt'g[e] as we do about that of d/g itself. The techniques used here are an extension of those of [H 2] which are in turn strongly related to those of [C; Q; V; W] and others. We begin by constructing several simplicial complexes from configurations of simple closed curves and properly imbedded arcs in a surface of genus g. The homology of the spin moduli space is identified with that of the spin mapping class group G, which acts on these complexes in a natural way. The Borel construction is then applied to obtain a spectral sequence which describes the homology of G in terms the homology of the stabilizers of the cells of these complexes. These turn out to be spin mapping class groups (in an extended sense) of smaller genus and the result is established inductively. The complexes are exactly the same as those of [H 2]; however, the spectral sequence arguments are more difficult because there are more orbits of cells under the action of G. Furthermore, in Sect. 4 we apply an entirely different and much simpler version of the argument of [H 2] to obtain stability in the case of a closed surface.