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arXiv · 2204.11696

The family signature theorem

Oscar Randal-Williams
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Abstract

We discuss several versions of the Family Signature Theorem: in rational cohomology using ideas of Meyer, in $KO[\tfrac{1}{2}]$-theory using ideas of Sullivan, and finally in symmetric $L$-theory using ideas of Ranicki. Employing recent developments in Grothendieck--Witt theory, we give a quite complete analysis of the resulting invariants. As an application we prove that the signature is multiplicative modulo 4 for fibrations of oriented Poincaré complexes, generalising a result of Hambleton, Korzeniewski and Ranicki, and discuss the multiplicativity of the de Rham invariant.

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Oscar Randal-Williams. 2022-12-09. The family signature theorem. https://doi.org/10.1017/prm.2022.91

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