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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

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.

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signature, fibre bundles, multiplicativity, Arf invariant, Brown–Kervaire invariant, modulo $8$
Mathematical Subject Classification 2010
Primary: 55R10, 55R12
Received: 25 January 2016
Revised: 13 August 2017
Accepted: 27 August 2017
Published: 3 April 2018
Carmen Rovi
Department of Mathematics
Indiana University
Bloomington, IN
United States