Birman-Craggs and Johnson homomorphisms

In this talk I will focus on the Torelli group, a subgroup of the mapping class group of a surface. For a start I will recall the Arf invariant and the Rochlin invariant. After that I will show how to construct a homology 3-sphere from an element of the Torelli group, so as to obtain certain homomorphisms from the Torelli group to $\mathbb Z/2\mathbb Z$. Finally we will find all homomorphisms to $\mathbb Z$ and by the Johnson theorem we compute the abelianization of the Torelli group.