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

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.

💡 Research Summary

The paper establishes a higher‑order Lefschetz fixed‑point formula for foliated manifolds in the presence of a closed Haefliger current. Starting from a smooth compact manifold M equipped with a regular foliation ℱ of codimension q, the author selects a closed Haefliger current ϕ∈H_c^(M/ℱ). By using Connes’ construction of the groupoid C‑algebra A=C*(M,ℱ) and the machinery of non‑commutative differential forms, ϕ is lifted to an equivariant cyclic cocycle τ_ϕ∈HC^G_{ev}(A), where G denotes the holonomy groupoid acting on A. The lift is performed via the Jaffe‑Lesniewski‑Osterwalder (JLO) chain and the Chern‑Connes character, ensuring that τ_ϕ is G‑invariant and well‑defined on the completed algebra.

Next, the author considers the G‑equivariant K‑theory K_*^G(A) and localizes it at a group element g∈G. The fixed‑point set Fix(g) consists of a finite union of submanifolds F, each transverse to the foliation and assumed to be elliptic (i.e., det(1−g|_{N_F})≠0). Using Kasparov’s KK‑theory and the Baum‑Connes assembly map, each K‑theory class