The family signature theorem

TL;DR

This paper explores various versions of the Family Signature Theorem across different cohomology theories, analyzing invariants and proving multiplicativity properties of signatures in fibrations of Poincaré complexes.

Contribution

It provides a comprehensive analysis of the Family Signature Theorem in multiple cohomology frameworks and extends multiplicativity results to new contexts.

Findings

Signature is multiplicative modulo 4 for fibrations of oriented Poincaré complexes.

Analyzes invariants using recent Grothendieck--Witt theory developments.

Discusses multiplicativity of the de Rham invariant.

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 Poincar\'e complexes, generalising a result of Hambleton, Korzeniewski and Ranicki, and discuss the multiplicativity of the de Rham invariant.