Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 14, Number 2 (2014), 627-669.
New series in the Johnson cokernels of the mapping class groups of surfaces
Let be a compact oriented surface of genus with one boundary component, and its mapping class group. Morita showed that the image of the Johnson homomorphism of is contained in the kernel of an –equivariant surjective homomorphism , where and is the degree part of the free Lie algebra generated by .
In this paper, we study the –module structure of the cokernel of the rational Johnson homomorphism , where . In particular, we show that the irreducible –module corresponding to a partition appears in the Johnson cokernel for any and with multiplicity one. We also give a new proof of the fact due to Morita that the irreducible –module corresponding to a partition appears in the Johnson cokernel with multiplicity one for odd .
The strategy of the paper is to give explicit descriptions of maximal vectors with highest weight and in the Johnson cokernel. Our construction is inspired by the Brauer–Schur–Weyl duality between and the Brauer algebras, and our previous work for the Johnson cokernel of the automorphism group of a free group.
Algebr. Geom. Topol., Volume 14, Number 2 (2014), 627-669.
Received: 20 August 2012
Revised: 3 August 2013
Accepted: 27 August 2013
First available in Project Euclid: 19 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20G05: Representation theory
Secondary: 57M50: Geometric structures on low-dimensional manifolds
Enomoto, Naoya; Satoh, Takao. New series in the Johnson cokernels of the mapping class groups of surfaces. Algebr. Geom. Topol. 14 (2014), no. 2, 627--669. doi:10.2140/agt.2014.14.627. https://projecteuclid.org/euclid.agt/1513715842