Localization of Basic Characteristic Classes

TL;DR

This paper introduces basic characteristic classes and numbers as new invariants for Riemannian foliations, linking their computation to the foliation’s dynamical behavior and providing an equivariant cohomology integration formula.

Contribution

It defines basic characteristic classes and numbers for Riemannian foliations and establishes a method to compute them using an Atiyah-Bott-Berline-Vergne-type formula.

Findings

Basic characteristic numbers are determined by the foliation's infinitesimal dynamics.

They can be computed via an equivariant basic cohomology integration formula.

The results apply to simply connected or transversely orientable Killing foliations.

Abstract

We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold M is simply connected (or more generally if the foliation is a transversely orientable Killing foliation), if M is complete and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type integration formula for equivariant basic cohomology.