The virtual cohomological dimension of the mapping class group of an orientable surface

Abstract

Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those of the arithmetic groups. If G is a linear algebraic group defined over Q, X is the symmetric space associated to G and A is an arithmetic subgroup of G, then A is virtually torsion free and acts properly discontinuously on X. The rational homology of A is the same as that of X/A. Furthermore there is a "bordification" of X ([BS]) to a manifold with corners 3~ and an extension of the action of A to a properly discontinuous action on Jf so that the quotient X / A is compact. The boundary of Jf is homotopy equivalent to a wedge of spheres, say of dimension d, and the virtual cohomological dimension of A is n d + 1, where n is the dimension of X. In the case of the mapping class group there is no analog for G, Ivanov (unpublished) has proven that F is not arithmetic. Nevertheless, F acts properly discontinuously on Teichmiiller space r which is homeomorphic to Euclidean space (of dimension 6 g 6 + 2 s ) ; 3"will play the role of the symmetric space. The quotient of .9by F is the moduli space of curves whose rational homology is then identified with that of E Harvey [Har] has constructed a Borel-Serre bordification J of 3-by analytic methods. In the case where F has punctures, we will build g by a different, combinatorial method. In addition, we will explicitly describe inside Y a cell complex Y of dimension 4 g 4 + s onto which J may be F-equivariantly retracted, thus establishing an analog of the constructions for SL n by Serre, Soul6 ([Sol) and Ash ([A]). This complex will be of the lowest possible dimension because we will use J= to prove our main result: