Singular Riemannian foliation
Final Report Abstract
The Poincaré-Hopf formula states that the Euler characteristic of a manifold is the sum of the indices of the zero set of a vector field. Similarly, Bott proved that Pontryagin numbers can be computed in terms of infinitesimal data around the zero set of a Killing field. These are two examples of theorems in which a topological invariant is computed in local terms. The goal of this project was to find a residue formula like this for the partition F of leaf closures of a Riemannian foliations F on M . This partition is a singular Riemannian foliation. As such it induces a natural stratification on the underlying manifold just as a Lie group action would do. The question is if the Pontryagin numbers of the normal bundle of F can be computed by infinitesimal data along a closed stratum. This is achieved for Killing foliations by relating the equivariant basic cohomology of F to the equivariant cohomology of an associated manifold. On the latter we can apply the Atiyah-Bott-Berline-Vergne fixed point theorem and transfer this to the equivariant basic cohomology. This gives a residue theorem for Killing foliations.
