Casson's invariant for homology 3-spheres and characteristic classes of surface bundles. I (Q1825478)

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.