The family signature theorem

Abstract

We discuss several versions of the Family Signature Theorem: in rational cohomology using ideas of Meyer, in KO[ 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, and discuss the multiplicativity of the de Rham invariant.