Higher Lefschetz formulas on Γ-proper manifolds

2025/12/16 by Paolo Piazza, Piazza, Paolo, Hessel Posthuma +5
#math.DG #math.KT

paper · pdf · doi:10.48550/arxiv.2512.14318

Abstract

Let Γ be a finitely generated discrete group acting properly and cocompactly on a smooth manifold M. By employing heat-kernel techniques we prove a geometric formula for the pairing of the index class associated to a Γ-equivariant Dirac operator D with a delocalized cyclic cocycles τ in HP^\bullet (ℂΓ,⟨ γ⟩). Our formula takes place on the fixed point manifold M^γ and should be regarded as a higher Lefschetz formula for D. The formula involves the Atiyah-Segal-Singer form and an explicit Z_γ-invariant form on M^γ that is naturally associated to τ∈ HP^\bullet (ℂΓ,⟨ γ⟩)

Related

• Primary and secondary invariants of Dirac operators on G-proper manifolds
• Orbifold index and equivariant K-homology
• Localized Index and L²-Lefschetz fixed point formula for orbifolds
• Higher orbital integrals, rho numbers and index theory
• Delocalized eta invariants of the signature operator on G-proper manifolds