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It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle F → E → B is multiplicative if the fundamental group π1(B) acts trivially on H∗(F; ℝ), with σ(E) = σ(F)σ(B). Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo 4, that is, σ(E) = σ(F)σ(B) mod 4. We present two results concerning the multiplicativity modulo 8: firstly we identify 1 4(σ(E) − σ(F)σ(B)) mod 2 with a ℤ2–valued Arf–Kervaire invariant of a Pontryagin squaring operation. Furthermore, we prove that if F is 2m–dimensional and the action of π1(B) is trivial on Hm(F, ℤ)∕torsion⊗ℤ4, this Arf–Kervaire invariant takes value 0 and hence the signature is multiplicative modulo 8, that is, σ(E) = σ(F)σ(B) mod 8.
Keywords: signature, fibre bundles, multiplicativity, Arf invariant, Brown–Kervaire invariant, modulo $8$
Rovi, Carmen 1
@article{10_2140_agt_2018_18_1281,
author = {Rovi, Carmen},
title = {The nonmultiplicativity of the signature modulo 8 of a fibre bundle is an {Arf{\textendash}Kervaire} invariant},
journal = {Algebraic and Geometric Topology},
pages = {1281--1322},
publisher = {mathdoc},
volume = {18},
number = {3},
year = {2018},
doi = {10.2140/agt.2018.18.1281},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1281/}
}
TY - JOUR AU - Rovi, Carmen TI - The nonmultiplicativity of the signature modulo 8 of a fibre bundle is an Arf–Kervaire invariant JO - Algebraic and Geometric Topology PY - 2018 SP - 1281 EP - 1322 VL - 18 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1281/ DO - 10.2140/agt.2018.18.1281 ID - 10_2140_agt_2018_18_1281 ER -
%0 Journal Article %A Rovi, Carmen %T The nonmultiplicativity of the signature modulo 8 of a fibre bundle is an Arf–Kervaire invariant %J Algebraic and Geometric Topology %D 2018 %P 1281-1322 %V 18 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1281/ %R 10.2140/agt.2018.18.1281 %F 10_2140_agt_2018_18_1281
Rovi, Carmen. The nonmultiplicativity of the signature modulo 8 of a fibre bundle is an Arf–Kervaire invariant. Algebraic and Geometric Topology, Tome 18 (2018) no. 3, pp. 1281-1322. doi: 10.2140/agt.2018.18.1281
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