[Groupe d'homéotopie de surfaces et invariant de Casson]
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.
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
@article{AIF_2004__54_4_1107_0,
author = {Perron, Bernard},
title = {Mapping class group and the {Casson} invariant},
journal = {Annales de l'Institut Fourier},
pages = {1107--1138},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {54},
number = {4},
year = {2004},
doi = {10.5802/aif.2045},
mrnumber = {2111023},
zbl = {02162421},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.2045/}
}
TY - JOUR AU - Perron, Bernard TI - Mapping class group and the Casson invariant JO - Annales de l'Institut Fourier PY - 2004 SP - 1107 EP - 1138 VL - 54 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2045/ DO - 10.5802/aif.2045 LA - en ID - AIF_2004__54_4_1107_0 ER -
%0 Journal Article %A Perron, Bernard %T Mapping class group and the Casson invariant %J Annales de l'Institut Fourier %D 2004 %P 1107-1138 %V 54 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2045/ %R 10.5802/aif.2045 %G en %F AIF_2004__54_4_1107_0
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] Braids, links and mapping class groups, Ann. of Math. Stud., vol 82, Princeton Univ. Press, Princeton, 1974 | MR | Zbl
[C] Lectures at MSRI (1985)
[G] Automorphisms of a 3-dimensional handlebody, Abh. Math. Sem. Univ. Hamburg, Volume 26 (1964), pp. 191-210 | MR | Zbl
[GG] Braids and Signatures (to appear in Bull. Math. Soc. France) | EuDML | Numdam | MR | Zbl
[GL] On the signature of a link, Invent. Math., Volume 47 (1978), pp. 53-69 | EuDML | MR | Zbl
[GM] Notes sur l'invariant de Casson des sphères d'homologie de dimension 3, Enseign. math., Volume 38 (1992), pp. 233-290 | MR | Zbl
[J1] An abelian quotient of the mapping class group , Math. Ann., Volume 249 (1980), pp. 225-242 | EuDML | MR | Zbl
[J2] The structure of the Torelli group I, Annals of Math., Volume 118 (1983), pp. 423-442 | MR | Zbl
[J3] The structure of the Torelli group II, Topology, Volume 24 (1985), pp. 113-126 | MR | Zbl
[Me1] Die signatur von lokalen koeffizientensystemen und Faserbündeln, Bonner Mathematische Schriften, Volume 53 (1972) | MR | Zbl
[Me2] Die signatur von Flächenbündeln, Math. Ann., Volume 201 (1973), pp. 239-264 | MR | Zbl
[Mo1] Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I, Topology, Volume 28 (1989), pp. 305-323 | MR | Zbl
[Mo2] On the structure of the Torelli group and the Casson invariant, Topology, Volume 30 (1991), pp. 603-621 | MR | Zbl
[Mo3] 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] 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] Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J., Volume 70 (1993), pp. 699-726 | MR | Zbl
[Mo6] 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] The generalized Casson invariant for 2-fold branched covers of and the Jones polynomial, Topology, Volume 32 (1993), pp. 419-438 | MR | Zbl
[Pe] Homomorphic extensions of Johnson homomorphisms via Fox Calculus, Ann. Inst. Fourier, Volume 54 (2004) no. 4, pp. 1073-1106 | Numdam | MR | Zbl
[Pi] 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] Two theorems on the mapping class group of surfaces, Proc. Amer. Math. Soc., Volume 68 (1978), pp. 347-350 | MR | Zbl
[PV] Groupe de monodromie géométrique des singularités simples, Math. Ann., Volume 306 (1996), pp. 231-245 | MR | Zbl
[R] Knots and links, Publish or Perish Inc., Berkeley CA, 1976 | MR | Zbl
[S] On homeomorphisms of a 3-dimensional handlebody, Can. J. Math., Volume 29 (1977), pp. 111-124 | MR | Zbl
[Si] Three dimensional manifolds and their Heegard diagrams, Trans. Amer. Math. Soc., Volume 35 (1933), pp. 88-111 | MR | Zbl
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