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

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Editorial Interests

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author Index

To Appear

Other MSP Journals

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)