#### Volume 14, issue 2 (2014)

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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
##### Abstract

Let ${\Sigma }_{g,1}$ be a compact oriented surface of genus $g$ with one boundary component, and ${\mathsc{ℳ}}_{g,1}$ its mapping class group. Morita showed that the image of the ${k}^{th}$ Johnson homomorphism ${\tau }_{k}^{\mathsc{ℳ}}$ of ${\mathsc{ℳ}}_{g,1}$ is contained in the kernel ${\mathfrak{h}}_{g,1}\left(k\right)$ of an $Sp$–equivariant surjective homomorphism $H{\otimes }_{ℤ}{\mathsc{ℒ}}_{2g}\left(k+1\right)\to {\mathsc{ℒ}}_{2g}\left(k+2\right)$, where $H:={H}_{1}\left({\Sigma }_{g,1},ℤ\right)$ and ${\mathsc{ℒ}}_{2g}\left(k\right)$ is the degree $k$ part of the free Lie algebra ${\mathsc{ℒ}}_{2g}$ generated by $H$.

In this paper, we study the $Sp$–module structure of the cokernel ${\mathfrak{h}}_{g,1}^{ℚ}\left(k\right)∕Im\left({\tau }_{k,ℚ}^{\mathsc{ℳ}}\right)$ of the rational Johnson homomorphism ${\tau }_{k,ℚ}^{\mathsc{ℳ}}:={\tau }_{k}^{\mathsc{ℳ}}\otimes {id}_{ℚ}$, where ${\mathfrak{h}}_{g,1}^{ℚ}\left(k\right):={\mathfrak{h}}_{g,1}\left(k\right){\otimes }_{ℤ}ℚ$. In particular, we show that the irreducible $Sp$–module corresponding to a partition $\left[{1}^{k}\right]$ appears in the ${k}^{th}$ Johnson cokernel for any $k\equiv 1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}4\right)$ and $k\ge 5$ with multiplicity one. We also give a new proof of the fact due to Morita that the irreducible $Sp$–module corresponding to a partition $\left[k\right]$ appears in the Johnson cokernel with multiplicity one for odd $k\ge 3$.

The strategy of the paper is to give explicit descriptions of maximal vectors with highest weight $\left[{1}^{k}\right]$ and $\left[k\right]$ in the Johnson cokernel. Our construction is inspired by the Brauer–Schur–Weyl duality between $Sp\left(2g,ℚ\right)$ 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
Primary: 20G05
Secondary: 57M50