The higher fixed point theorem for foliations I. Holonomy invariant currents

TL;DR

This paper develops a higher Lefschetz formula for foliations using holonomy invariant currents, linking cyclic cohomology and K-theory to generalize classical fixed point results.

Contribution

It introduces a novel approach to fixed point formulas for foliations via Haefliger currents and cyclic cohomology, extending classical theorems.

Findings

Derived a higher Lefschetz formula for foliations

Connected cyclic cohomology with fixed point contributions

Generalized classical fixed point results using new currents

Abstract

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. We associate with such a current an equivariant cyclic cohomology class of Connes' C*-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points. As special cases, we recover a number of classical results, and since we may use any closed Haefliger current on the foliation, we get new and very interesting formulae.