### Abstract

Original language | English |
---|---|

Pages (from-to) | 4069–4091 |

Number of pages | 23 |

Journal | Algebraic & Geometric Topology |

Volume | 18 |

DOIs | |

Publication status | Published - 11 Dec 2018 |

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### Keywords

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

### Cite this

*Algebraic & Geometric Topology*,

*18*, 4069–4091. https://doi.org/10.2140/agt.2018.18.4069

**Cohomology of symplectic groups and Meyer’s signature theorem.** / Benson, Dave; Campagnolo, Caterina; Ranicki, Andrew; Rovi, Carmen.

Research output: Contribution to journal › Article

*Algebraic & Geometric Topology*, vol. 18, pp. 4069–4091. https://doi.org/10.2140/agt.2018.18.4069

}

TY - JOUR

T1 - Cohomology of symplectic groups and Meyer’s signature theorem

AU - Benson, Dave

AU - Campagnolo, Caterina

AU - Ranicki, Andrew

AU - Rovi, Carmen

N1 - Campagnolo acknowledges support by the Swiss National Science Foundation, grant number PP00P2-128309/1, and by the German Science Foundation via the Research Training Group 2229, under which this research was started and then completed.

PY - 2018/12/11

Y1 - 2018/12/11

N2 - 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.

AB - 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.

KW - surface bundles

KW - signature modulo 8

KW - signature cocycle

KW - Meyer

KW - group cohomology

KW - symplectic groups

U2 - 10.2140/agt.2018.18.4069

DO - 10.2140/agt.2018.18.4069

M3 - Article

VL - 18

SP - 4069

EP - 4091

JO - Algebraic & Geometric Topology

JF - Algebraic & Geometric Topology

SN - 1472-2747

ER -