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 ! SpC.2g; Z/ ! Sp.2g; Z/ ! 1 the pullback of the universal cover of CSp.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 2 2gC1. There is a central extension 1 ! Z=2 ! Hz ! H ! 1, and Hz appears as a quotient of the metaplectic double cover Mp.2g; Z/ D SpC.2g; Z/=2Z. It is an extension of Sp.2g; 2/ by an almost extraspecial group of order 2 2gC2 , and has a faithful irreducible complex representation of dimension 2g. Provided g > 4, the extension Hz 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.