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.
Keywords
mapping class group, Johnson homomorphism, Torelli group