Casson's invariant for homology 3-spheres and characteristic classes of surface bundles
Let \(H_ g\) be a 3-dimensional handle-body of genus g and \(S_ g=\partial H_ g\), a closed surface of genus g. The Torelli group \(T_{g,1}\) of \(S_ g\), relative an embedded disc D 2 in \(S_ g\), is the group of mapping classes of \(S_ g\) relative D 2 inducing the identity on homology. Given an element \(\phi\) in \(T_{g,1}\), there is a canonical way to glue together two copies of \(H_ g\) along their boundaries to produce (a Heegaard splitting of) a homology 3-sphere M(\(\phi)\), so the Casson invariant \(\lambda\) (M(\(\phi)\))\(\in {\mathbb{Z}}\) is defined and gives a map \(\lambda\) : \(T_{g,1}\to {\mathbb{Z}}\). ``The purpose of the present note is to announce our result concerning the map \(\lambda\). Briefly speaking we have shown that the Casson invariant is a kind of secondary invariant associated with the characteristic classes of surface bundles introduced in several papers of the author. As a result we have obtained an alternative description of \(\lambda\).''
characteristic classes of surface bundles
scientific article; zbMATH DE number 4044759