Partially defined cocycles and the Maslov index for a local ring

Amedeo Mazzoleni

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Homomorphic extensions of Johnson homomorphisms via Fox calculus

Bernard Perron (2004)

Annales de l’institut Fourier

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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 $k + 1 t h$ 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.