#### Volume 7, issue 3 (2007)

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

### Matthew B Day

Algebraic & Geometric Topology 7 (2007) 1297–1326
 arXiv: math.GT/0702127
##### 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 $k\ge 2$, we construct a crossed homomorphism ${ϵ}_{k}$ which extends Morita’s homomorphism ${\stackrel{̃}{\tau }}_{k}$ to the entire mapping class group. From this crossed homomorphism we also obtain a crossed homomorphism extending the $k$th Johnson homomorphism ${\tau }_{k}$ 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.

##### Keywords
mapping class group, Johnson homomorphism, Torelli group
Primary: 57N05
Secondary: 57T15