Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 18, Number 3 (2018), 1281-1322.
The nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invariant
It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group acts trivially on , with . Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo , that is, . We present two results concerning the multiplicativity modulo : firstly we identify with a –valued Arf–Kervaire invariant of a Pontryagin squaring operation. Furthermore, we prove that if is –dimensional and the action of is trivial on , this Arf–Kervaire invariant takes value and hence the signature is multiplicative modulo , that is, .
Algebr. Geom. Topol., Volume 18, Number 3 (2018), 1281-1322.
Received: 25 January 2016
Revised: 13 August 2017
Accepted: 27 August 2017
First available in Project Euclid: 26 April 2018
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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