Algebraic & Geometric Topology

New series in the Johnson cokernels of the mapping class groups of surfaces

Naoya Enomoto and Takao Satoh

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Let Σg,1 be a compact oriented surface of genus g with one boundary component, and g,1 its mapping class group. Morita showed that the image of the kth Johnson homomorphism τk of g,1 is contained in the kernel hg,1(k) of an Sp–equivariant surjective homomorphism H2g(k+1)2g(k+2), where H:=H1(Σg,1,) and 2g(k) is the degree k part of the free Lie algebra 2g generated by H.

In this paper, we study the Sp–module structure of the cokernel hg,1(k)Im(τk,) of the rational Johnson homomorphism τk,:=τk id, where hg,1(k):=hg,1(k). In particular, we show that the irreducible Sp–module corresponding to a partition [1k] appears in the kth Johnson cokernel for any k1(mod4) and k5 with multiplicity one. We also give a new proof of the fact due to Morita that the irreducible Sp–module corresponding to a partition [k] appears in the Johnson cokernel with multiplicity one for odd k3.

The strategy of the paper is to give explicit descriptions of maximal vectors with highest weight [1k] and [k] in the Johnson cokernel. Our construction is inspired by the Brauer–Schur–Weyl duality between Sp(2g,) and the Brauer algebras, and our previous work for the Johnson cokernel of the automorphism group of a free group.

Article information

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20G05: Representation theory
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Johnson homomorphism mapping class group


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.

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