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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

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

surface bundles, signature modulo 8, signature cocycle, Meyer, group cohomology, symplectic groups
Mathematical Subject Classification 2010
Primary: 20J06
Secondary: 20C33, 55R10
Received: 26 November 2017
Revised: 30 May 2018
Accepted: 16 June 2018
Published: 11 December 2018
Dave Benson
Institute of Mathematics
University of Aberdeen
United Kingdom
Caterina Campagnolo
Department of Mathematics
Karlsruhe Institute of Technology
Andrew Ranicki
School of Mathematics
University of Edinburgh
United Kingdom
Carmen Rovi
Department of Mathematics
Indiana University Bloomington
Bloomington, IN
United States