Homomorphic extensions of Johnson homomorphisms via Fox calculus

Perron, Bernard

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 ${ℳ}_{g,1}$ d’une surface de genre $g$ et de bord à une composante, qui coïncide avec le $k+1^{\text{ème}}$ 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 ${ℳ}_{g,1}$

Using Fox differential calculus, for any positive integer $k$, we construct a map on the mapping class group ${ℳ}_{g,1}$ 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 ${ℳ}_{g,1}$ of the second and third Johnson-Morita homomorphisms.

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

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