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2018 The nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invariant
Carmen Rovi
Algebr. Geom. Topol. 18(3): 1281-1322 (2018). DOI: 10.2140/agt.2018.18.1281

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 .

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Carmen Rovi. "The nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invariant." Algebr. Geom. Topol. 18 (3) 1281 - 1322, 2018. https://doi.org/10.2140/agt.2018.18.1281

Information

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

zbMATH: 06866400
MathSciNet: MR3784006
Digital Object Identifier: 10.2140/agt.2018.18.1281

Subjects:
Primary: 55R10, 55R12

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 3 • 2018
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