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

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 ${\pi }_1\left(B\right)$ acts trivially on $H^{\ast }\left(F;ℝ\right)$, with $\sigma \left(E\right)=\sigma \left(F\right)\sigma \left(B\right)$. Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo $4$, that is, $\sigma \left(E\right)=\sigma \left(F\right)\sigma \left(B\right)mod4$. We present two results concerning the multiplicativity modulo $8$: firstly we identify $\frac{1}{4}\left(\sigma \left(E\right)-\sigma \left(F\right)\sigma \left(B\right)\right)mod2$ with a ${\mathbb{Z}}_2$–valued Arf–Kervaire invariant of a Pontryagin squaring operation. Furthermore, we prove that if $F$ is $2m$–dimensional and the action of ${\pi }_1\left(B\right)$ is trivial on $H^m\left(F,\mathbb{Z}\right)∕torsion\otimes {\mathbb{Z}}_4$, this Arf–Kervaire invariant takes value $0$ and hence the signature is multiplicative modulo $8$, that is, $\sigma \left(E\right)=\sigma \left(F\right)\sigma \left(B\right)mod8$.