Algebraic & Geometric Topology

The nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invariant

Carmen Rovi

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Abstract

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle F E B is multiplicative if the fundamental group π 1 ( B ) acts trivially on H ( F ; ) , with σ ( E ) = σ ( F ) σ ( B ) . Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo 4 , that is, σ ( E ) = σ ( F ) σ ( B ) mod 4 . We present two results concerning the multiplicativity modulo 8 : firstly we identify 1 4 ( σ ( E ) σ ( F ) σ ( B ) ) mod 2 with a 2 –valued Arf–Kervaire invariant of a Pontryagin squaring operation. Furthermore, we prove that if F is 2 m –dimensional and the action of π 1 ( B ) is trivial on H m ( F , ) torsion 4 , this Arf–Kervaire invariant takes value 0 and hence the signature is multiplicative modulo 8 , that is, σ ( E ) = σ ( F ) σ ( B ) mod 8 .

Article information

Source
Algebr. Geom. Topol., Volume 18, Number 3 (2018), 1281-1322.

Dates
Received: 25 January 2016
Revised: 13 August 2017
Accepted: 27 August 2017
First available in Project Euclid: 26 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1524708093

Digital Object Identifier
doi:10.2140/agt.2018.18.1281

Mathematical Reviews number (MathSciNet)
MR3784006

Zentralblatt MATH identifier
06866400

Subjects
Primary: 55R10: Fiber bundles 55R12: Transfer

Keywords
signature fibre bundles multiplicativity Arf invariant Brown–Kervaire invariant modulo $8$

Citation

Rovi, Carmen. The nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invariant. Algebr. Geom. Topol. 18 (2018), no. 3, 1281--1322. doi:10.2140/agt.2018.18.1281. https://projecteuclid.org/euclid.agt/1524708093


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