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On the structure of the Torelli group and the Casson invariant. (English) Zbl 0747.57010
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.

57N10 Topology of general \(3\)-manifolds (MSC2010)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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