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2007 Extending Johnson's and Morita's homomorphisms to the mapping class group
Matthew B Day
Algebr. Geom. Topol. 7(3): 1297-1326 (2007). DOI: 10.2140/agt.2007.7.1297


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 k2, 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.


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Matthew B Day. "Extending Johnson's and Morita's homomorphisms to the mapping class group." Algebr. Geom. Topol. 7 (3) 1297 - 1326, 2007.


Received: 26 February 2007; Revised: 3 August 2007; Accepted: 15 August 2007; Published: 2007
First available in Project Euclid: 20 December 2017

zbMATH: 1181.57025
MathSciNet: MR2350283
Digital Object Identifier: 10.2140/agt.2007.7.1297

Primary: 57N05
Secondary: 57T15

Rights: Copyright © 2007 Mathematical Sciences Publishers


Vol.7 • No. 3 • 2007
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