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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


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 2, we construct a crossed homomorphism ϵk which extends Morita’s homomorphism τ̃k to the entire mapping class group. From this crossed homomorphism we also obtain a crossed homomorphism extending the kth Johnson homomorphism τk 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.

mapping class group, Johnson homomorphism, Torelli group
Mathematical Subject Classification 2000
Primary: 57N05
Secondary: 57T15
Received: 26 February 2007
Revised: 3 August 2007
Accepted: 15 August 2007
Published: 24 September 2007
Matthew B Day
Department of Mathematics
The University of Chicago
5734 South University Avenue
Chicago IL 60637