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 , we construct a crossed homomorphism which extends Morita’s homomorphism to the entire mapping class group. From this crossed homomorphism we also obtain a crossed homomorphism extending the th Johnson homomorphism 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.
"Extending Johnson's and Morita's homomorphisms to the mapping class group." Algebr. Geom. Topol. 7 (3) 1297 - 1326, 2007. https://doi.org/10.2140/agt.2007.7.1297