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

Day, Matthew B¹

¹Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago IL 60637, USA

Algebraic and Geometric Topology, Tome 7 (2007) no. 3, pp. 1297-1326

Résumé

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.

Détail

DOI : 10.2140/agt.2007.7.1297

Keywords: mapping class group, Johnson homomorphism, Torelli group

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