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It was proved by Chern, Hirzebruch and Serre that the signature of a fibrebundle is multiplicative if the fundamental group of the base acts trivially onthe cohomology ring of the fibre with real coefficients, in which case thesignature of the total space equals the product of the signatures of base andfibre. Hambleton, Korzeniewski and Ranicki proved that in any case thesignature is multiplicative modulo 4. In this paper we present two resultsconcerning the multiplicativity modulo 8: firstly we identify the obstructionto multiplicativity modulo 8 with the Arf-Kervaire invariant of a Pontryaginsquaring operation. Furthermore, we prove that if the fibre is even-dimensionaland the action of the fundamental group of the base is trivial on the middlecohomology of the fibre with $\mathbb{Z}_4$ coefficients, then thisArf-Kervaire invariant takes value 0 and hence the signature is multiplicativemodulo 8.

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