Volume 7, issue 3 (2007)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 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.

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