The non-multiplicativity of the signature modulo 8 of a fibre bundle is an Arf-Kervaire invariant

TL;DR

This paper investigates the conditions under which the signature of a fibre bundle is multiplicative modulo 8, linking the obstruction to the Arf-Kervaire invariant and identifying cases where multiplicativity holds.

Contribution

It identifies the Arf-Kervaire invariant as the obstruction to signature multiplicativity modulo 8 and establishes conditions for when this invariant vanishes, ensuring multiplicativity.

Findings

Obstruction to multiplicativity modulo 8 is the Arf-Kervaire invariant.

If the fibre is even-dimensional and the action on middle cohomology is trivial mod 4, the invariant is zero.

Under these conditions, the signature is multiplicative modulo 8.

Abstract

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group of the base acts trivially on the cohomology ring of the fibre with real coefficients, in which case the signature of the total space equals the product of the signatures of base and fibre. Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo 4. In this paper we present two results concerning the multiplicativity modulo 8: firstly we identify the obstruction to multiplicativity modulo 8 with the Arf-Kervaire invariant of a Pontryagin squaring operation. Furthermore, we prove that if the fibre is even-dimensional and the action of the fundamental group of the base is trivial on the middle cohomology of the fibre with $\mathbb{Z}_4$ coefficients, then this Arf-Kervaire invariant takes value 0 and hence the signature is multiplicative modulo 8.