Cohomology of symplectic groups and Meyer's signature theorem

Abstract

Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of $4$, and can be computed using an element of $H^2\left(\mathsf{Sp}\left(2g,\mathbb{Z}\right),\mathbb{Z}\right)$. If we denote by $1\to \mathbb{Z}\to \widetilde{\mathsf{Sp}\left(2g,\mathbb{Z}\right)}\to \mathsf{Sp}\left(2g,\mathbb{Z}\right)\to 1$ the pullback of the universal cover of $\mathsf{Sp}\left(2g,ℝ\right)$, then by a theorem of Deligne, every finite index subgroup of $\widetilde{\mathsf{Sp}\left(2g,\mathbb{Z}\right)}$ contains $2\mathbb{Z}$. As a consequence, a class in the second cohomology of any finite quotient of $\mathsf{Sp}\left(2g,\mathbb{Z}\right)$ can at most enable us to compute the signature of a surface bundle modulo $8$. We show that this is in fact possible and investigate the smallest quotient of $\mathsf{Sp}\left(2g,\mathbb{Z}\right)$ that contains this information. This quotient $\mathfrak{ℌ}$ is a nonsplit extension of $\mathsf{Sp}\left(2g,2\right)$ by an elementary abelian group of order $2^{2g+1}$. There is a central extension $1\to \mathbb{Z}∕2\to \widetilde{\mathfrak{ℌ}}\to \mathfrak{ℌ}\to 1$, and $\widetilde{\mathfrak{ℌ}}$ appears as a quotient of the metaplectic double cover $\mathsf{Mp}\left(2g,\mathbb{Z}\right)=\widetilde{\mathsf{Sp}\left(2g,\mathbb{Z}\right)}∕2\mathbb{Z}$. It is an extension of $\mathsf{Sp}\left(2g,2\right)$ by an almost extraspecial group of order $2^{2g+2}$, and has a faithful irreducible complex representation of dimension $2^g$. Provided $g⩾4$, the extension $\widetilde{\mathfrak{ℌ}}$ is the universal central extension of $\mathfrak{ℌ}$. Putting all this together, in Section 4 we provide a recipe for computing the signature modulo $8$, and indicate some consequences.