Higher Lefschetz formulas on {\Gamma}-proper manifolds

TL;DR

This paper establishes a higher Lefschetz formula for Dirac operators on manifolds with a proper cocompact group action, linking index theory, cyclic cohomology, and fixed point formulas using heat kernel methods.

Contribution

It provides a geometric formula for the pairing of the index class with delocalized cyclic cocycles, extending Lefschetz formulas to higher and noncommutative settings.

Findings

Derived a fixed point formula involving Atiyah-Segal-Singer form

Connected index pairing with cyclic cocycles to geometric data

Extended Lefschetz formulas to higher and noncommutative contexts

Abstract

Let $\Gamma$ 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 $\Gamma$-equivariant Dirac operator $D$ with a delocalized cyclic cocycles $\tau$ in $HP^\bullet (\mathbb{C}\Gamma,\langle \gamma \rangle)$. Our formula takes place on the fixed point manifold $M^\gamma$ and should be regarded as a higher Lefschetz formula for $D$. The formula involves the Atiyah-Segal-Singer form and an explicit $Z_\gamma$-invariant form on $M^\gamma$ that is naturally associated to $\tau\in HP^\bullet (\mathbb{C}\Gamma,\langle \gamma \rangle)$