Volume 18, issue 3 (2018)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 18
Issue 5, 2509–3131
Issue 4, 1883–2507
Issue 3, 1259–1881
Issue 2, 635–1258
Issue 1, 1–633

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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
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 π1(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 2m–dimensional and the action of π1(B) is trivial on Hm(F, )torsion4, 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