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Extending Johnson's and Morita's homomorphisms to the
mapping class group
Matthew B Day
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Algebraic & Geometric Topology 7
(2007) 1297–1326
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Abstract
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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
~τk 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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Keywords
mapping class group, Johnson
homomorphism, Torelli group
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Mathematical Subject Classification
Primary: 57N05
Secondary: 57T15
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Publication
Received: 26 February 2007
Revised: 3 August 2007
Accepted: 15 August 2007
Published: 24 September 2007
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