Cohomology of symplectic groups and Meyer's signature theorem

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 H2(Sp(2g,ℤ),ℤ). If we denote by 1→ℤ→Sp(2g,ℤ)˜→Sp(2g,ℤ)→1 the pullback of the universal cover of Sp(2g,ℝ), then by a theorem of Deligne, every finite index subgroup of Sp(2g,ℤ)˜ contains 2ℤ. As a consequence, a class in the second cohomology of any finite quotient of Sp(2g,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 Sp(2g,ℤ) that contains this information. This quotient ℌ is a nonsplit extension of Sp(2g,2) by an elementary abelian group of order 22g+1. There is a central extension 1→ℤ∕2→ℌ→ℌ→1, and ℌ appears as a quotient of the metaplectic double cover Mp(2g,ℤ)=Sp(2g,ℤ)˜∕2ℤ. It is an extension of Sp(2g,2) by an almost extraspecial group of order 22g+2, and has a faithful irreducible complex representation of dimension 2g. Provided g⩾4, the extension ℌ is the universal central extension of ℌ. Putting all this together, in Section 4 we provide a recipe for computing the signature modulo 8, and indicate some consequences.