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 $H^2(\mathsf{Sp}(2g,\mathbb{Z}),\mathbb{Z})$. Denoting by $1\to \mathbb{Z}\to \widetilde{\mathsf{Sp}(2g,\mathbb{Z})}\to \mathsf{Sp}(2g,\mathbb{Z})\to 1$ the pullback of the universal cover of $\mathsf{Sp}(2g,\mathbb{R})$, Deligne proved that every finite index subgroup of $\widetilde{\mathsf{Sp}(2g,\mathbb{Z})}$ contains $2\mathbb{Z}$. As a consequence, a class in the second cohomology of any finite quotient of $\mathsf{Sp}(2g,\mathbb{Z})$ 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 $\mathsf{Sp}(2g,\mathbb{Z})$ that contains this information. This quotient $\mathfrak{H}$ is a non-split extension of $\mathsf{Sp}(2g,2)$ by an elementary abelian group of order $2^{2g+1}$. There is a central extension $1\to \mathbb{Z}/2\to \tilde{\mathfrak{H}}\to \mathfrak{H}\to 1$, and $\tilde{\mathfrak{H}}$ appears as a quotient of the metaplectic double cover $\mathsf{Mp}(2g,\mathbb{Z})=\widetilde{\mathsf{Sp}(2g,\mathbb{Z})}/2\mathbb{Z}$. It is an extension of $\mathsf{Sp}(2g,2)$ by an almost extraspecial group of order $2^{2g+2}$, and has a faithful irreducible complex representation of dimension $2^g$. Provided $g\ge 4$, $\tilde{\mathfrak{H}}$ is the universal central extension of $\mathfrak{H}$. Putting all this together, we provide a recipe for computing the signature modulo $8$, and indicate some consequences.

18 pages. Minor corrections. The most important one is in the table for $g=1$ on page 16: two columns had been swapped in the previous version. This is the version accepted for publication in Algebraic and Geometric Topology

References in corpus (2)

• Finite quotients of symplectic groups vs mapping class groups
• The non-multiplicativity of the signature modulo 8 of a fibre bundle is an Arf-Kervaire invariant

Cited by in corpus (2)

• Anomalies of Duality Groups and Extended Conformal Manifolds
• Mapping class group of manifolds which look like $3$-dimensional complete intersections