On the structure of the Torelli group and the Casson invariant (Q1184327)

The Torelli group is the subgroup of the mapping class group of a surface acting trivially on the first homology of the surface. Reglueing the standard Heegaard splitting of genus \(g>1\) of the 3-sphere \(S^ 3\) by an element of the Torelli group gives a homology 3-sphere so that the Casson invariant is defined and gives a mapping of the Torelli group to the integers. In a previous paper studying this mapping, the author found a strong connection between the Casson invariant of a homology 3-sphere and his theory of characteristic classes of surface bundles developed in a series of papers. These results are extended in the present paper to the more general situation of an embedding of a surface (not necessarily Heegaard) into an arbitrary homology 3-sphere. Again such an embedding defines a mapping of the Torelli group to the integers, which is the main object of study of the present paper, e.g., how much it differs from a homomorphism. On the way, the algebraic structure of a certain quotient of the Torelli group is studied which measures how the elements of the Torelli group act on the fourth nilpotent quotient of the fundamental group of the surface.