Cohomology of symplectic groups and Meyer’s signature theorem

Benson, Dave; Campagnolo, Caterina; Ranicki, Andrew; Rovi, Carmen

doi:10.2140/agt.2018.18.4069

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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

DOI: 10.2140/agt.2018.18.4069

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 $H^{2} (Sp (2 g, ℤ), ℤ)$ . If we denote by $1 \to ℤ \to \tilde{Sp (2 g, ℤ)} \to Sp (2 g, ℤ) \to 1$ the pullback of the universal cover of $Sp (2 g, ℝ)$ , then by a theorem of Deligne, every finite index subgroup of $\tilde{Sp (2 g, ℤ)}$ contains $2 ℤ$ . As a consequence, a class in the second cohomology of any finite quotient of $Sp (2 g, ℤ)$ 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 $Sp (2 g, ℤ)$ that contains this information. This quotient $ℌ$ is a nonsplit extension of $Sp (2 g, 2)$ by an elementary abelian group of order $2^{2 g + 1}$ . There is a central extension $1 \to ℤ ∕ 2 \to \tilde{ℌ} \to ℌ \to 1$ , and $\tilde{ℌ}$ appears as a quotient of the metaplectic double cover $Mp (2 g, ℤ) = \tilde{Sp (2 g, ℤ)} ∕ 2 ℤ$ . It is an extension of $Sp (2 g, 2)$ by an almost extraspecial group of order $2^{2 g + 2}$ , and has a faithful irreducible complex representation of dimension $2^{g}$ . Provided $g \geq 4$ , the extension $\tilde{ℌ}$ 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.

Keywords

surface bundles, signature modulo 8, signature cocycle, Meyer, group cohomology, symplectic groups

Mathematical Subject Classification 2010

Primary: 20J06

Secondary: 20C33, 55R10

References

Publication

Received: 26 November 2017

Revised: 30 May 2018

Accepted: 16 June 2018

Published: 11 December 2018

Authors

Dave Benson
Institute of Mathematics University of Aberdeen Aberdeen United Kingdom

Caterina Campagnolo
Department of Mathematics Karlsruhe Institute of Technology Karlsruhe Germany

Andrew Ranicki
School of Mathematics University of Edinburgh Edinburgh United Kingdom

Carmen Rovi
Department of Mathematics Indiana University Bloomington Bloomington, IN United States

Volume 18, issue 7 (2018)