Equivariant homology for pseudo-differential operators

TL;DR

This paper computes the cyclic homology of cross-product algebras of symbols on manifolds with finite group actions, linking it to de Rham cohomology of fixed point sets, and provides new insights into such homologies.

Contribution

It introduces a spectral sequence approach to identify cyclic homology of cross-product algebras with de Rham cohomology of fixed point manifolds, including explicit cases.

Findings

Identifies cyclic homology with de Rham cohomology of fixed point manifolds.

Provides explicit homology computations for $C^{ ext{infty}}(M) times ext{ extGamma}$.

Develops new results on homologies of general cross-product algebras.

Abstract

We compute the cyclic homology for the cross-product al- gebra $A(M)\rtimes\Gamma$ of the algebra of complete symbols on a compact man- ifold $M$ with action of a finite group $\Gamma$. A spectral sequence argument shows that these groups can be identified using deRham cohomology of the fixed point manifolds $S ^*M ^g$ . In the process we obtain new re- sults about the homologies of general cross-product algebras and provide explicit identification of the homologies for $C^{\infty}(M)\rtimes \Gamma$.