The Unstable Integral Homology of the Mapping Class Groups of a Surface with Boundary

We construct a graph complex calculating the integral homology of the bordered mapping class groups. We compute the homology of the bordered mapping class groups of the surfaces S1,1, S1,2 and S2,1. Using the circle action on this graph complex, we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups. We compute the homology of the punctured mapping class groups associated to the surfaces S1,1 and S2,1. Finally, we use Miller’s operad to get the first Kudo-Araki and Browder operations on our graph complex. We also consider an unstable version of the higher Kudo-Araki-Dyer-Lashoff operations.