#### Volume 18, issue 7 (2018)

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Cohomology of symplectic groups and Meyer's signature theorem

### Dave Benson, Caterina Campagnolo, Andrew Ranicki and Carmen Rovi

Algebraic & Geometric Topology 18 (2018) 4069–4091
##### 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 ${H}^{2}\left(\mathsf{Sp}\left(2g,ℤ\right),ℤ\right)$. If we denote by $1\to ℤ\to \stackrel{˜}{\mathsf{Sp}\left(2g,ℤ\right)}\to \mathsf{Sp}\left(2g,ℤ\right)\to 1$ the pullback of the universal cover of $\mathsf{Sp}\left(2g,ℝ\right)$, then by a theorem of Deligne, every finite index subgroup of $\stackrel{˜}{\mathsf{Sp}\left(2g,ℤ\right)}$ contains $2ℤ$. As a consequence, a class in the second cohomology of any finite quotient of $\mathsf{Sp}\left(2g,ℤ\right)$ 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 $\mathsf{Sp}\left(2g,ℤ\right)$ 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 $1\to ℤ∕2\to \stackrel{̃}{\mathfrak{ℌ}}\to \mathfrak{ℌ}\to 1$, and $\stackrel{̃}{\mathfrak{ℌ}}$ appears as a quotient of the metaplectic double cover $\mathsf{Mp}\left(2g,ℤ\right)=\stackrel{˜}{\mathsf{Sp}\left(2g,ℤ\right)}∕2ℤ$. 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 $g\ge 4$, the extension $\stackrel{̃}{\mathfrak{ℌ}}$ 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.

##### Keywords
surface bundles, signature modulo 8, signature cocycle, Meyer, group cohomology, symplectic groups
##### Mathematical Subject Classification 2010
Primary: 20J06
Secondary: 20C33, 55R10