It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group acts trivially on , with . Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo , that is, . We present two results concerning the multiplicativity modulo : firstly we identify with a –valued Arf–Kervaire invariant of a Pontryagin squaring operation. Furthermore, we prove that if is –dimensional and the action of is trivial on , this Arf–Kervaire invariant takes value and hence the signature is multiplicative modulo , that is, .
"The nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invariant." Algebr. Geom. Topol. 18 (3) 1281 - 1322, 2018. https://doi.org/10.2140/agt.2018.18.1281