Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 18, Number 7 (2018), 4069-4091.
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 , and can be computed using an element of . If we denote by the pullback of the universal cover of , then by a theorem of Deligne, every finite index subgroup of contains . As a consequence, a class in the second cohomology of any finite quotient of can at most enable us to compute the signature of a surface bundle modulo . We show that this is in fact possible and investigate the smallest quotient of that contains this information. This quotient is a nonsplit extension of by an elementary abelian group of order . There is a central extension , and appears as a quotient of the metaplectic double cover . It is an extension of by an almost extraspecial group of order , and has a faithful irreducible complex representation of dimension . Provided , the extension is the universal central extension of . Putting all this together, in Section 4 we provide a recipe for computing the signature modulo , and indicate some consequences.
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4069-4091.
Received: 26 November 2017
Revised: 30 May 2018
Accepted: 16 June 2018
First available in Project Euclid: 18 December 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20J06: Cohomology of groups
Secondary: 20C33: Representations of finite groups of Lie type 55R10: Fiber bundles
Benson, Dave; Campagnolo, Caterina; Ranicki, Andrew; Rovi, Carmen. Cohomology of symplectic groups and Meyer's signature theorem. Algebr. Geom. Topol. 18 (2018), no. 7, 4069--4091. doi:10.2140/agt.2018.18.4069. https://projecteuclid.org/euclid.agt/1545102064