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

Enomoto, Naoya; Satoh, Takao

doi:10.2140/agt.2014.14.627

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

Naoya Enomoto and Takao Satoh

Algebraic & Geometric Topology 14 (2014) 627–669

DOI: 10.2140/agt.2014.14.627

Abstract

Let $Σ_{g, 1}$ be a compact oriented surface of genus $g$ with one boundary component, and $ℳ_{g, 1}$ its mapping class group. Morita showed that the image of the $k^{t h}$ Johnson homomorphism $τ_{k}^{ℳ}$ of $ℳ_{g, 1}$ is contained in the kernel $h_{g, 1} (k)$ of an $Sp$ –equivariant surjective homomorphism $H \otimes_{ℤ} ℒ_{2 g} (k + 1) \to ℒ_{2 g} (k + 2)$ , where $H : = H_{1} (Σ_{g, 1}, ℤ)$ and $ℒ_{2 g} (k)$ is the degree $k$ part of the free Lie algebra $ℒ_{2 g}$ generated by $H$ .

In this paper, we study the $Sp$ –module structure of the cokernel $h_{g, 1}^{ℚ} (k) ∕ Im (τ_{k, ℚ}^{ℳ})$ of the rational Johnson homomorphism $τ_{k, ℚ}^{ℳ} : = τ_{k}^{ℳ} \otimes {id}_{ℚ}$ , where $h_{g, 1}^{ℚ} (k) : = h_{g, 1} (k) \otimes_{ℤ} ℚ$ . In particular, we show that the irreducible $Sp$ –module corresponding to a partition $[1^{k}]$ appears in the $k^{t h}$ Johnson cokernel for any $k \equiv 1 (mod 4)$ and $k \geq 5$ 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 $k \geq 3$ .

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

Dedicated to the memory of Midori Kato

Keywords

Johnson homomorphism, mapping class group

Mathematical Subject Classification 2010

Primary: 20G05

Secondary: 57M50

References

Publication

Received: 20 August 2012

Revised: 3 August 2013

Accepted: 27 August 2013

Published: 30 January 2014

Authors

Naoya Enomoto
Department of Mathematics Faculty of Science Nara Women’s University Kitauoyahigashi-machi Nara city 630-8506 Japan

Takao Satoh
Department of Mathematics Faculty of Science Division II Tokyo University of Science Kagurazaka 1-3 Shinjuku-ku Tokyo 1628601 Japan

Volume 14, issue 2 (2014)