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

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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Keywords

mapping class group, Johnson homomorphism, Torelli group

Mathematical Subject Classification 2000

Primary: 57N05

Secondary: 57T15

References

Publication

Received: 26 February 2007

Revised: 3 August 2007

Accepted: 15 August 2007

Published: 24 September 2007

Authors

Matthew B Day
Department of Mathematics The University of Chicago 5734 South University Avenue Chicago IL 60637 USA

Volume 7, issue 3 (2007)

Matthew B Day

Abstract