\title{Cohomology of symplectic groups and Meyer's signature theorem}

\author{Dave Benson}
\address{Institute of Mathematics, University of Aberdeen, Aberdeen
  AB24 3UE, Scotland, United Kingdom}
\author{Caterina Campagnolo}
\address{Department of Mathematics, Karlsruhe Institute of Technology,
D-76128 Karlsruhe, Germany}
\author{Andrew Ranicki}
\address{School of Mathematics, University of Edinburgh, Edinburgh EH9
  3FD, Scotland, United Kingdom}
\author{Carmen Rovi}
\address{Department of Mathematics, Indiana University, Bloomington IN 47405, USA}

\begin{document}

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

\subjclass[2010]{20J06 (primary); 55R10, 20C33 (secondary)}