Excision in Hochschild and cyclic homology without continuous linear sections

TL;DR

This paper proves excision properties for Hochschild and cyclic homology in certain topological algebras, enabling explicit computations for Whitney and smooth function algebras on closed subsets of manifolds.

Contribution

It establishes excision results for Hochschild and cyclic homology without requiring continuous linear sections, extending their computability to Whitney and smooth function algebras.

Findings

Hochschild and cyclic homology satisfy excision for nuclear H-unital Fréchet algebras

Explicit computation of these homologies for Whitney functions on closed subsets

Periodic cyclic homology computed for smooth and Whitney function algebras

Abstract

We prove that continuous Hochschild and cyclic homology satisfy excision for extensions of nuclear H-unital Frechet algebras and use this to compute them for the algebra of Whitney functions on an arbitrary closed subset of a smooth manifold. Using a similar excision result for periodic cyclic homology, we also compute the periodic cyclic homology of algebras of smooth functions and Whitney functions on closed subsets of smooth manifolds.