Homomorphic extensions of Johnson homomorphisms via Fox calculus
[Extensions homomorphes des homomorphismes de Johnson via le calcul de Fox]
Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 1073-1106.

A l’aide du calcul différentiel de Fox, on définit pour tout entier positif k, une application sur le groupe d’homéotopie g,1 d’une surface de genre g et de bord à une composante, qui coïncide avec le k+1 ème homomorphisme de Johnson- Morita quand on la restreint à un sous-groupe approprié. Ceci permet d’obtenir de façon très simple une extension homomorphe des deuxième et troisième homomorphismes de Johnson- Morita à tout le groupe g,1

Using Fox differential calculus, for any positive integer k, we construct a map on the mapping class group g,1 of a surface of genus g with one boundary component, such that, when restricted to an appropriate subgroup, it coincides with the k+1th Johnson-Morita homomorphism. This allows us to construct very easily a homomorphic extension to g,1 of the second and third Johnson-Morita homomorphisms.

DOI : 10.5802/aif.2044
Classification : 57M05
Keywords: mapping class group of a surface, Johnson-Morita homomorphisms, Fox differential calculus
Mot clés : groupe d'homéotopie d'une surface, homomorphismes de Johnson-Morita, calcul différentiel de Fox
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. Homomorphic extensions of Johnson homomorphisms via Fox calculus. Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 1073-1106. doi : 10.5802/aif.2044. http://archive.numdam.org/articles/10.5802/aif.2044/

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

[Br] K. Brown Cohomology of groups, Graduate Texts in Math., 87, Springer-Verlag, 1982 | MR | Zbl

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

[F] R. Fox Free differential calculus I, Annals of Math., Volume 57 (1953), pp. 547-560 | MR | Zbl

[H] S. Humphries Generators of the mapping class group (Lecture Notes in Math.), Volume 722 (1979), pp. 44-47 | 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

[KMS] A. Karass; W. Magnus; D. Solitar Combinatorial group theory, Pure Appl. Math., 13, Interscience Publ., New York, 1966 | 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 (Contemporary 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

[Pe] B. Perron Mapping class group and the Casson invariant, Ann. Inst. Fourier, Volume 54 (2004) no. 4, pp. 1107-1138 | Numdam | MR | Zbl

[Po] J. Powell Two theorems on the mapping class group of surfaces, Proc. AMS, 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

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

[Sa] N. Saveliev Lectures on the topology of 3-manifolds. An introduction to the Casson Invariant, De Gruyter Text Book, Berlin, New York, 1999 | MR | Zbl

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