Higher Lefschetz formulas on Γ-proper manifolds

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 $HP^\bullet (\mathbb{C}Γ,\langle γ\rangle)$. 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 $τ\in HP^\bullet (\mathbb{C}Γ,\langle γ\rangle)$

💡 Research Summary

The paper “Higher Lefschetz formulas on Γ‑proper manifolds” addresses the problem of extending Lefschetz‑type fixed‑point formulas to higher cyclic cocycles in the setting of a non‑compact manifold M on which a finitely generated discrete group Γ acts properly and cocompactly. The authors consider a Γ‑equivariant Dirac operator D acting on a Z₂‑graded Γ‑vector bundle E over M. The classical Atiyah‑Segal‑Singer fixed‑point theorem gives a pointwise formula for the equivariant index of D at a group element g when Γ is a compact Lie group. In the non‑compact, discrete‑group case the index class lives in the K‑theory of the algebra of Γ‑compactly supported smoothing operators A_c^Γ(M,E). Pairing this K‑class with the canonical trace Tr_Γ reproduces Atiyah’s Γ‑index, which coincides with the ordinary index on the compact quotient M/Γ.

Beyond the trace, one can pair the index class with higher cyclic cocycles τ ∈ HC^⋆(ℂΓ). The cyclic cohomology of the group algebra splits into contributions from the identity conjugacy class and from non‑trivial conjugacy classes ⟨γ⟩. For τ supported on the identity, Connes‑Moscovici’s higher index theorem provides a formula involving the A‑hat class of M/Γ and the pull‑back of a class β(τ) from the classifying space BΓ. For τ supported on a non‑trivial class, earlier work (Wang‑Wang) gave a “zero‑order” Lefschetz formula: \