A new definition of the Casson invariant
In a previous paper, following work of \textit{S. Morita} [Topology 28, No. 3, 305--323 (1989; Zbl 0684.57008)], the author defined an invariant \(\delta (f)\) for any \(f\) in the mapping class group of a compact oriented surface with one boundary component, and showed, using the surgery formula for Casson's invariant of homology 3-spheres, that if \(f\) is an element of a certain subgroup of the Torelli group (the subgroup of the mapping class group acting trivially on homology), then \(\delta (f)\) coincides with the Casson invariant of the homology 3-sphere \(M_f\) obtained by gluing two handlebodies along their boundaries using \(f\) (also, every homology 3-sphere is obtained in this way). In the present paper, it is shown directly that \(\delta (f)\) depends only on \(M_f\) (i.e. without reference to the Casson invariant), by showing that \(\delta (f)\) is invariant under the two Reidemeister-Singer operations for Heegaard splittings (stabilization and equivalence). This requires some ``lengthy, but elementary'' computations; on the other hand, the surgery formula, which is a difficult point in Casson's approach, follows almost immediately from the definition of \(\delta (f)\).
scientific article; zbMATH DE number 1744378