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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 →ℌ̃ → ℌ → 1, and ℌ̃ 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 ℌ̃ 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
Benson, Dave 1 ; Campagnolo, Caterina 2 ; Ranicki, Andrew 3 ; Rovi, Carmen 4
@article{10_2140_agt_2018_18_4069,
author = {Benson, Dave and Campagnolo, Caterina and Ranicki, Andrew and Rovi, Carmen},
title = {Cohomology of symplectic groups and {Meyer{\textquoteright}s} signature theorem},
journal = {Algebraic and Geometric Topology},
pages = {4069--4091},
publisher = {mathdoc},
volume = {18},
number = {7},
year = {2018},
doi = {10.2140/agt.2018.18.4069},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.4069/}
}
TY - JOUR AU - Benson, Dave AU - Campagnolo, Caterina AU - Ranicki, Andrew AU - Rovi, Carmen TI - Cohomology of symplectic groups and Meyer’s signature theorem JO - Algebraic and Geometric Topology PY - 2018 SP - 4069 EP - 4091 VL - 18 IS - 7 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.4069/ DO - 10.2140/agt.2018.18.4069 ID - 10_2140_agt_2018_18_4069 ER -
%0 Journal Article %A Benson, Dave %A Campagnolo, Caterina %A Ranicki, Andrew %A Rovi, Carmen %T Cohomology of symplectic groups and Meyer’s signature theorem %J Algebraic and Geometric Topology %D 2018 %P 4069-4091 %V 18 %N 7 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.4069/ %R 10.2140/agt.2018.18.4069 %F 10_2140_agt_2018_18_4069
Benson, Dave; Campagnolo, Caterina; Ranicki, Andrew; Rovi, Carmen. Cohomology of symplectic groups and Meyer’s signature theorem. Algebraic and Geometric Topology, Tome 18 (2018) no. 7, pp. 4069-4091. doi: 10.2140/agt.2018.18.4069
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