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

Abstract

Let ${\Sigma }_{g,1}$ be a compact oriented surface of genus $g$ with one boundary component, and ${\mathcal{ℳ}}_{g,1}$ its mapping class group. Morita showed that the image of the $k^{th}$ Johnson homomorphism ${\tau }_k^{\mathcal{ℳ}}$ of ${\mathcal{ℳ}}_{g,1}$ is contained in the kernel ${\mathfrak{h}}_{g,1}\left(k\right)$ of an $Sp$–equivariant surjective homomorphism $H{\otimes }_{\mathbb{Z}}{\mathcal{ℒ}}_{2g}\left(k+1\right)\to {\mathcal{ℒ}}_{2g}\left(k+2\right)$, where $H:=H_1\left({\Sigma }_{g,1},\mathbb{Z}\right)$ and ${\mathcal{ℒ}}_{2g}\left(k\right)$ is the degree $k$ part of the free Lie algebra ${\mathcal{ℒ}}_{2g}$ generated by $H$.

In this paper, we study the $Sp$–module structure of the cokernel ${\mathfrak{h}}_{g,1}^{\mathbb{Q}}\left(k\right)∕Im\left({\tau }_{k,\mathbb{Q}}^{\mathcal{ℳ}}\right)$ of the rational Johnson homomorphism ${\tau }_{k,\mathbb{Q}}^{\mathcal{ℳ}}:={\tau }_k^{\mathcal{ℳ}}\otimes {id}_{\mathbb{Q}}$, where ${\mathfrak{h}}_{g,1}^{\mathbb{Q}}\left(k\right):={\mathfrak{h}}_{g,1}\left(k\right){\otimes }_{\mathbb{Z}}\mathbb{Q}$. 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 $k\equiv 1\left(mod4\right)$ 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 $Sp\left(2g,\mathbb{Q}\right)$ and the Brauer algebras, and our previous work for the Johnson cokernel of the automorphism group of a free group.