The family signature theorem

Hirzebruch's signature theorem relates the signature of the intersection form of a manifold with the integral over the manifold of a certain characteristic class, namely the L-class. This was extended to families of smooth manifolds (i.e. smooth fibre bundles) by Atiyah, using the family index theorem for the fibrewise signature operator. In this setting it relates the Chern character of a certain vector bundle constructed from the local system of intersection forms of the fibres, with the fibre integral of the L-class.

I will explain an elaboration of this result in two directions: to families more general than smooth fibre bundles (e.g. topological fibre bundles), and to an equation in symmetric L-theory rather than rational cohomology. I will then say something about how the L-theoretic version can be analysed, using recent advances in Grothendieck--Witt theory.