Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 7, Number 3 (2007), 1297-1326.
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 , we construct a crossed homomorphism which extends Morita’s homomorphism to the entire mapping class group. From this crossed homomorphism we also obtain a crossed homomorphism extending the th Johnson homomorphism 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.
Algebr. Geom. Topol., Volume 7, Number 3 (2007), 1297-1326.
Received: 26 February 2007
Revised: 3 August 2007
Accepted: 15 August 2007
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57N05: Topology of $E^2$ , 2-manifolds
Secondary: 57T15: Homology and cohomology of homogeneous spaces of Lie groups
Day, Matthew B. Extending Johnson's and Morita's homomorphisms to the mapping class group. Algebr. Geom. Topol. 7 (2007), no. 3, 1297--1326. doi:10.2140/agt.2007.7.1297. https://projecteuclid.org/euclid.agt/1513796742