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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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##### References:
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