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 S _1,1, S _1,2 and S _2,1. 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 S _1,1 and S _2,1. 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.