The family signature theorem

Abstract

We discuss several versions of the Family Signature Theorem: in rational cohomology using ideas of Meyer, in $KO[\frac{1}{2}]$-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.

34 pages; v2 36 pages, accepted version to appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics (Ranicki memorial volume)

References in corpus (1)

• Algebraic independence of topological Pontryagin classes

Cited by in corpus (1)

• Some rational homology computations for diffeomorphisms of odd-dimensional manifolds