Index theory for basic Dirac operators on Riemannian foliations

Abstract

In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs.

33 pages

References in corpus (3)

• Modified differentials and basic cohomology for Riemannian foliations
• The invertible double of elliptic operators
• The eta invariant and equivariant index of transversally elliptic operators

Cited by in corpus (3)

• Analysis on singular spaces: Lie manifolds and operator algebras
• Cohomological localization for transverse Lie algebra actions on Riemannian foliations
• Symmetric Polynomials in Tropical Algebra Semirings