zbMATH — the first resource for mathematics

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

57N10 Topology of general \(3\)-manifolds (MSC2010)
57R20 Characteristic classes and numbers in differential topology
55R10 Fiber bundles in algebraic topology
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.