Mapping class group and the Casson invariant
[Groupe d'homéotopie de surfaces et invariant de Casson]
Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 1107-1138.

En utilisant une nouvelle définition des second et troisième homomorphismes de Johnson, on simplifie et on généralise les résultats de Morita sur l'invariant de Casson des sphères d'homologie définies par scindement de Heegard. En particulier, on calcule l'invariant de Casson des sphères d'homologie obtenues en recollant deux corps d'anses par un homéomorphisme du groupe de Torelli.

Using a new definition of the second and third Johsnon homomorphisms, we simplify and extend the work of Morita on the Casson invariant of homology-spheres defined by Heegard splittings. In particular, we calculate the Casson invariant of the homology-sphere obtained by gluing two handlebodies along a homeomorphism of the boundary belonging to the Torelli subgroup.

DOI : 10.5802/aif.2045
Classification : 57M05
Keywords: mapping class group, Johnson-Morita homomorphisms, homology spheres, Casson invariant
Mot clés : groupe d'homéotopie, homomorphismes de Johnson-Morita, sphères d'homologie, invariant de Casson
Perron, Bernard 1

1 Université de Bourgogne, Institut de mathématiques de Bourgogne, UFR sciences et techniques, 9 avenue Alain Savary, BP 47870, 21078 Dijon cedex (France)
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Perron, Bernard. Mapping class group and the Casson invariant. Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 1107-1138. doi : 10.5802/aif.2045. http://archive.numdam.org/articles/10.5802/aif.2045/

[B] J. Birman Braids, links and mapping class groups, Ann. of Math. Stud., vol 82, Princeton Univ. Press, Princeton, 1974 | MR | Zbl

[C] A. Casson Lectures at MSRI (1985)

[G] H.B. Griffiths Automorphisms of a 3-dimensional handlebody, Abh. Math. Sem. Univ. Hamburg, Volume 26 (1964), pp. 191-210 | MR | Zbl

[GG] J.-M. Gambaudo; E. Ghys Braids and Signatures (to appear in Bull. Math. Soc. France) | EuDML | Numdam | MR | Zbl

[GL] C.M.cA. Gordon; R. Litherland On the signature of a link, Invent. Math., Volume 47 (1978), pp. 53-69 | EuDML | MR | Zbl

[GM] L. Guillou; A. Marin Notes sur l'invariant de Casson des sphères d'homologie de dimension 3, Enseign. math., Volume 38 (1992), pp. 233-290 | MR | Zbl

[J1] D. Johnson An abelian quotient of the mapping class group g , Math. Ann., Volume 249 (1980), pp. 225-242 | EuDML | MR | Zbl

[J2] D. Johnson The structure of the Torelli group I, Annals of Math., Volume 118 (1983), pp. 423-442 | MR | Zbl

[J3] D. Johnson The structure of the Torelli group II, Topology, Volume 24 (1985), pp. 113-126 | MR | Zbl

[Me1] W. Meyer Die signatur von lokalen koeffizientensystemen und Faserbündeln, Bonner Mathematische Schriften, Volume 53 (1972) | MR | Zbl

[Me2] W. Meyer Die signatur von Flächenbündeln, Math. Ann., Volume 201 (1973), pp. 239-264 | MR | Zbl

[Mo1] S. Morita Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I, Topology, Volume 28 (1989), pp. 305-323 | MR | Zbl

[Mo2] S. Morita On the structure of the Torelli group and the Casson invariant, Topology, Volume 30 (1991), pp. 603-621 | MR | Zbl

[Mo3] S. Morita The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math., Volume 111 (1993), pp. 197-224 | MR | Zbl

[Mo4] S. Morita The structure of the mapping class group and characteristic classes of surface bundles, Mapping class groups and Moduli spaces of Riemann surfaces (Contemp. Math.), Volume 150 (1993), pp. 303-315 | Zbl

[Mo5] S. Morita Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J., Volume 70 (1993), pp. 699-726 | MR | Zbl

[Mo6] S. Morita Casson invariant, signature defect of framed manifolds and second characteristic classes of surface bundles, J. Diff. Geom., Volume 47 (1997), pp. 560-599 | MR | Zbl

[Mu] D. Mullins The generalized Casson invariant for 2-fold branched covers of § 3 and the Jones polynomial, Topology, Volume 32 (1993), pp. 419-438 | MR | Zbl

[Pe] B. Perron Homomorphic extensions of Johnson homomorphisms via Fox Calculus, Ann. Inst. Fourier, Volume 54 (2004) no. 4, pp. 1073-1106 | Numdam | MR | Zbl

[Pi] W. Pitsch Une construction intrinsèque du cÏur de l'invariant de Casson, Ann. Inst. Fourier, Volume 51 (2001) no. 6, pp. 1741-1761 | Numdam | MR | Zbl

[Po] J. Powell Two theorems on the mapping class group of surfaces, Proc. Amer. Math. Soc., Volume 68 (1978), pp. 347-350 | MR | Zbl

[PV] B. Perron; J.-P. Vannier Groupe de monodromie géométrique des singularités simples, Math. Ann., Volume 306 (1996), pp. 231-245 | MR | Zbl

[R] D. Rolfsen Knots and links, Publish or Perish Inc., Berkeley CA, 1976 | MR | Zbl

[S] S. Suzuky On homeomorphisms of a 3-dimensional handlebody, Can. J. Math., Volume 29 (1977), pp. 111-124 | MR | Zbl

[Si] J. Singer Three dimensional manifolds and their Heegard diagrams, Trans. Amer. Math. Soc., Volume 35 (1933), pp. 88-111 | MR | Zbl

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