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 : https://doi.org/10.5802/aif.2045
Classification : 57M05
Mots 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'institut Fourier},
     volume = {54},
     number = {4},
     year = {2004},
     doi = {10.5802/aif.2045},
     zbl = {02162421},
     mrnumber = {2111023},
     language = {en},
     url = {www.numdam.org/item/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://www.numdam.org/item/AIF_2004__54_4_1107_0/

[B] J. Birman Braids, links and mapping class groups, Ann. of Math. Stud., Volume vol 82, Princeton Univ. Press, Princeton, 1974 | MR 375281 | Zbl 0305.57013

[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 159313 | Zbl 0229.57005

[GG] J.-M. Gambaudo; E. Ghys Braids and Signatures (to appear in Bull. Math. Soc. France) | EuDML 272312 | Numdam | MR 2233695 | Zbl 1103.57001

[GL] C.M.cA. Gordon; R. Litherland On the signature of a link, Invent. Math., Volume 47 (1978), pp. 53-69 | EuDML 142569 | MR 500905 | Zbl 0391.57004

[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 1189008 | Zbl 0776.57008

[J1] D. Johnson An abelian quotient of the mapping class group g , Math. Ann., Volume 249 (1980), pp. 225-242 | EuDML 163403 | MR 579103 | Zbl 0409.57009

[J2] D. Johnson The structure of the Torelli group I, Annals of Math., Volume 118 (1983), pp. 423-442 | MR 727699 | Zbl 0549.57006

[J3] D. Johnson The structure of the Torelli group II, Topology, Volume 24 (1985), pp. 113-126 | MR 793178 | Zbl 0571.57009

[Me1] W. Meyer Die signatur von lokalen koeffizientensystemen und Faserbündeln, Bonner Mathematische Schriften, Volume 53 (1972) | MR 305402 | Zbl 0243.58004

[Me2] W. Meyer Die signatur von Flächenbündeln, Math. Ann., Volume 201 (1973), pp. 239-264 | MR 331382 | Zbl 0241.55019

[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 1014464 | Zbl 0684.57008

[Mo2] S. Morita On the structure of the Torelli group and the Casson invariant, Topology, Volume 30 (1991), pp. 603-621 | MR 1133875 | Zbl 0747.57010

[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 1193604 | Zbl 0787.57008

[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 0791.57018

[Mo5] S. Morita Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J., Volume 70 (1993), pp. 699-726 | MR 1224104 | Zbl 0801.57011

[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 1617632 | Zbl 0903.57008

[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 1217078 | Zbl 0784.57003

[Pe] B. Perron Homomorphic extensions of Johnson homomorphisms via Fox Calculus, Ann. Inst. Fourier, Volume 54 (2004) no. 4, pp. 1073-1106 | Numdam | MR 2111022 | Zbl 02162420

[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 1871288 | Zbl 0983.57010

[Po] J. Powell Two theorems on the mapping class group of surfaces, Proc. Amer. Math. Soc., Volume 68 (1978), pp. 347-350 | MR 494115 | Zbl 0391.57009

[PV] B. Perron; J.-P. Vannier Groupe de monodromie géométrique des singularités simples, Math. Ann., Volume 306 (1996), pp. 231-245 | MR 1411346 | Zbl 0863.32013

[R] D. Rolfsen Knots and links, Publish or Perish Inc., Berkeley CA, 1976 | MR 515288 | Zbl 0339.55004

[S] S. Suzuky On homeomorphisms of a 3-dimensional handlebody, Can. J. Math., Volume 29 (1977), pp. 111-124 | MR 433433 | Zbl 0339.57001

[Si] J. Singer Three dimensional manifolds and their Heegard diagrams, Trans. Amer. Math. Soc., Volume 35 (1933), pp. 88-111 | MR 1501673 | Zbl 0006.18501