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Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles. (English) Zbl 0641.57004
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
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[1] J. Birman: On Siegel’s modular group. Math. Ann., 191, 59-68 (1971). · Zbl 0208.10601 · doi:10.1007/BF01433472 · eudml:162122
[2] J. Birman: Braids, links, and mapping class groups. Ann. Math. Studies, 82, Princeton Univ. Press (1975). · Zbl 0305.57013
[3] J. Birman and R. Craggs: The ^-invariant of 3-manif olds and certain structural properties of the group of homeomorphisms of a closed oriented 2-manifold. Trans. Amer. Math. Soc, 237, 283-309 (1978). · Zbl 0383.57006 · doi:10.2307/1997623
[4] D. Johnson: An abelian quotient of the mapping class group Sg. Math. Ann., 249, 225-242 (1980). · Zbl 0409.57009 · doi:10.1007/BF01363897 · eudml:163403
[5] D. Johnson: A survey of the Torelli group. Contemporary Math., 20, 165-179 (1983). · Zbl 0553.57002 · doi:10.1090/conm/020/718141
[6] D. Johnson: The structure of the Torelli group, II and III. Topology,24,113-144 (1985). · Zbl 0571.57010 · doi:10.1016/0040-9383(85)90050-3
[7] S. Morita: Characteristic classes of surface bundles. Bull. Amer. Math. Soc, 11, 386-388 (1984). · Zbl 0579.55006 · doi:10.1090/S0273-0979-1984-15321-7
[8] S. Morita: Families of Jacobian manifolds and characteristic classes of surface bundles, I and II (preprint). · Zbl 0672.57015 · doi:10.5802/aif.1188 · numdam:AIF_1989__39_3_777_0 · eudml:74852
[9] S. Morita: Casson’s invariant for homology 3-spheres and the mapping class group. Proc Japan Acad., 62A, 402-405 (1986). · Zbl 0623.57006 · doi:10.3792/pjaa.62.402
[10] S. Suzuki: On homeomorphisms of a 3-dimensional handlebody. Can. J. Math., 29, 111-124 (1977). · Zbl 0339.57001 · doi:10.4153/CJM-1977-011-1
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