On the abelianization of the Torelli group

Abstract

Around 1980 Dennis Johnson computed the abelianized Torelli group for surfaces of genus at least three and at most one boundary component. His computation shows that it is completely described in terms of the first homology group of the surface. We show in this thesis that this result holds for all surfaces with boundary and genus at least three. Our method differs from his, is inductive in nature and may open the way to compute also higher homology groups of the Torelli group. Further, we compute the Torelli group for low genus in some cases, using the result of Mess for genus two.