Global Homotopies for Differential Hochschild Cohomologies

TL;DR

This paper develops explicit global homotopies for differential Hochschild cochains, enhancing the Hochschild-Kostant-Rosenberg map and enabling new computations in Hochschild cohomology within differential geometry.

Contribution

It introduces a novel method combining symbol calculus and coalgebraic techniques to construct deformation retracts in various geometric settings.

Findings

Recovered the classical Hochschild-Kostant-Rosenberg theorem.

Computed previously inaccessible Hochschild cohomologies.

Developed deformation retracts for principal bundles and invariant contexts.

Abstract

We construct explicit global homotopies for differential Hochschild cochains in differential geometry, thereby upgrading the classical Hochschild-Kostant-Rosenberg map to a deformation retract. Our approach combines two key techniques: a symbol calculus from differential geometry and a coalgebraic version of the van Est theorem. To demonstrate its effectiveness, we develop deformation retracts in several related settings, including principal bundles and invariant contexts. As a byproduct, we recover the classical Hochschild-Kostant-Rosenberg theorem and compute previously inaccessible Hochschild cohomologies.