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Extending Johnson's and Morita's homomorphisms to the mapping class group

Matthew B. Day
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Abstract

We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every $k\geq 2$, we construct a crossed homomorphism $ε_k$ which extends Morita's homomorphism $\tilde τ_k$ to the entire mapping class group. From this crossed homomorphism we also obtain a crossed homomorphism extending the $k$th Johnson homomorphism $τ_k$ to the mapping class group. D. Johnson and S. Morita obtained their respective homomorphisms by considering the action of the mapping class group on the nilpotent truncations of the surface group; our approach is to mimic Morita's construction topologically by using nilmanifolds associated to these truncations. This allows us to take the ranges of these crossed homomorphisms to be certain finite-dimensional real vector spaces associated to these nilmanifolds.

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BibTeXRIS

Matthew B. Day. 2007-10-01. Extending Johnson's and Morita's homomorphisms to the mapping class group. https://doi.org/10.2140/agt.2007.7.1297

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