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The nonmultiplicativity of the signature modulo 8 of a fibre bundle is an Arf-Kervaire invariant. (English) Zbl 1393.55008
The signature of a fibre bundle \(F\rightarrow E\rightarrow B\) is multiplicative if the fundamental group \(\pi_1(B)\) acts trivially on \(H^\ast(F;\mathbb R)\). It is further known that in any case the signature is multiplicative modulo \(4\). In this paper the author shows that \(\frac{1}{4}(\sigma(E)-\sigma(F)\sigma(B)) \) mod \(2\) can be identified with a \(\mathbb Z_2\)-valued Arf-Kervaire invariant of a Pontrjyagin squaring operation. Further if \(F\) is \(2m\)-dimensional and the action of \(\pi_1(B)\) is trivial on \(H^m(F;\mathbb Z)/\mathrm{ torsion }\otimes\mathbb Z_4\), this Arf-invariant is zero and the signature is multiplicative mod \(8\). The key feature of the argument is the construction of a model for the chain complex of the total space that gives enough information to compute its signature. The model is inspired by the transfer map in quadratic \(L\)-theory of Lück and Ranicki.

MSC:
55R10 Fiber bundles in algebraic topology
55R12 Transfer for fiber spaces and bundles in algebraic topology
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