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Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles. I. (English) Zbl 0684.57008

This is a profound study of the relationship between the Casson invariant of a homology-3-sphere and the structure of certain subgroups of the mapping-class group of a closed surface related to the Torelli group, i.e. the subgroup of mapping classes acting trivially on the homology of the surface (and giving rise, via Heegaard splittings, to homology 3- spheres). It makes use of D. Johnson’s study of the Torelli group and an interpretation of the Casson invariant as a secondary invariant associated with the characteristic classes of surface bundles over surfaces. An integer-valued function on the mapping-class group is defined, and the main result of the paper states that its restriction to the subgroup of the Torelli group generated by twists along bounding simple-closed curves (which is sufficient to construct, via Heegaard splittings, all homology-3-spheres) is essentially Casson’s invariant of the corresponding homology-3-spheres.
Reviewer: B.Zimmermann

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57R50 Differential topological aspects of diffeomorphisms
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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