Casson's invariant for homology 3-spheres and the mapping class group (Q1091636)

In 1978 \textit{J. S. Birman} and \textit{R. Craggs} [Trans. Am. Math. Soc. 237, 283--309 (1978; Zbl 0383.57006)] introduced \(\mathbb{Z}/2\)-invariants of homeomorphisms of surfaces, making use of the Rokhlin invariants of homology 3-spheres. The paper under review is considered with a refinement of Birman-Craggs invariants based on the Casson integer invariant \(\lambda\) of oriented homology 3-spheres. Let \(S\) be the closed orientable surface of genus \(\geq 2\), and let \(S_0=S\setminus \operatorname{Int} D\) where \(D\) is a disk in \(S\). Let \(E\) be the set of all embeddings \(S\hookrightarrow S^3\). Let \(G_1,G_2,\ldots\), \(G_{n+1}=[G_n,G_1],\ldots\) be the lower central series of \(\pi_1(S_0)=G_1\). Let \(M_n\) be the subgroup of the mapping class group of \(S_0\) consisting of the elements which act trivially on \(G_1/G_n\). The author defines a mapping \(\Lambda: M_2\to \operatorname{Map}(E, \mathbb{Z})\): If \(\phi \in M_2\) and \(f\in E\) then \(\Lambda (\phi)(f)=\lambda (W_{\phi})\), where \(W_{\phi}\) is the oriented homology 3-sphere obtained by cutting \(S^3\) along \(f(S)\) and regluing back the resulting two pieces by the extension of \(\phi\) to \(S\). Several theorems concerned with computation and properties of \(\Lambda\) are announced. In particular, the value \(\Lambda(\phi)\) is stated to depend only on the class of \(\phi\) in \(M_2/M_5\). This implies that it is possible to define the Casson invariant of homology 3-spheres in terms of the action of the pasting maps of Heegaard decompositions on the fifth nilpotent quotient of the fundamental group of the Heegaard surface.