#### Volume 18, issue 3 (2018)

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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
##### 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, $\sigma \left(E\right)=\sigma \left(F\right)\sigma \left(B\right)\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}4$. We present two results concerning the multiplicativity modulo $8$: firstly we identify $\frac{1}{4}\left(\sigma \left(E\right)-\sigma \left(F\right)\sigma \left(B\right)\right)\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}2$ with a ${ℤ}_{2}$–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 ${H}^{m}\left(F,ℤ\right)∕torsion\otimes {ℤ}_{4}$, this Arf–Kervaire invariant takes value $0$ and hence the signature is multiplicative modulo $8$, that is, $\sigma \left(E\right)=\sigma \left(F\right)\sigma \left(B\right)\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}8$.

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signature, fibre bundles, multiplicativity, Arf invariant, Brown–Kervaire invariant, modulo $8$