Mod 2 homology of the stable spin mapping class group

Abstract

We compute the mod 2 homology of spin mapping class groups in the stable range. In earlier work we computed the stable mod p homology of the oriented mapping class group, and the methods and results here are very similar. The forgetful map from the spin mapping class group to the oriented mapping class groups induces a homology isomorphism for odd p but for p=2 it is far from being an isomorphism. We include a general discussion of tangential structures on 2-manifolds and their mapping class groups and then specialise to spin structures. As in Madsen and Weiss' solution to Mumford's conjecture, the stable homology is the homology of the zero space of a certain Thom spectrum.