Algebraic & Geometric Topology

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

Matthew B Day

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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 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.

Article information

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

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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

mapping class group Johnson homomorphism Torelli group


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.

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