Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles. (English) Zbl 0641.57004

Morita, Shigeyuki

Proc. Japan Acad., Ser. A 63, 229-232 (1987).

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 \(\lambda\).”

Reviewer: B.Zimmermann

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57R20	Characteristic classes and numbers in differential topology
55R10	Fiber bundles in algebraic topology

Keywords:

3-dimensional handle-body; Torelli group; group of mapping classes; homology 3-sphere; Casson invariant; characteristic classes of surface bundles

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