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

Rovi, Carmen

doi:10.2140/agt.2018.18.1281

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

Carmen Rovi

Algebraic & Geometric Topology 18 (2018) 1281–1322

DOI: 10.2140/agt.2018.18.1281

Abstract

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle $F \to E \to B$ is multiplicative if the fundamental group $π_{1} (B)$ acts trivially on $H^{*} (F; ℝ)$ , 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 $\frac{1}{4} (σ (E) - σ (F) σ (B)) mod 2$ with a $ℤ_{2}$ –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 $H^{m} (F, ℤ) ∕ torsion \otimes ℤ_{4}$ , this Arf–Kervaire invariant takes value $0$ and hence the signature is multiplicative modulo $8$ , that is, $σ (E) = σ (F) σ (B) mod 8$ .

Keywords

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

Mathematical Subject Classification 2010

Primary: 55R10, 55R12

References

Publication

Received: 25 January 2016

Revised: 13 August 2017

Accepted: 27 August 2017

Published: 3 April 2018

Authors

Carmen Rovi
Department of Mathematics Indiana University Bloomington, IN United States
http://pages.iu.edu/~crovi/index.html

Volume 18, issue 3 (2018)