On piecewise smooth cohomology of Lie groupoids and Lie algebroids

TL;DR

This paper explores the relationship between piecewise smooth cohomology of Lie groupoids and Lie algebroids, leveraging Mishchenko's theorem to connect different cohomological theories and their applications.

Contribution

It establishes two applications of Mishchenko's theorem, linking piecewise de Rham cohomology of Lie groupoids with cohomology of Lie algebroids and combining invariant cohomology results.

Findings

Demonstrates isomorphism between piecewise smooth and Lie algebroid cohomology for transitive Lie algebroids

Relates piecewise de Rham cohomology of Lie groupoids to Lie algebroid cohomology

Integrates classical invariant cohomology results with Mishchenko's theorem

Abstract

Mishchenko's theorem states that piecewise smooth and Lie algebroid cohomology of a transitive Lie algebroid defined over a combinatorial manifold are isomorphic. In this paper, we describe two applications of that result. The first application consists in the relationship between piecewise de Rham cohomology of a Lie groupoid and piecewise smooth cohomology of its Lie algebroid. For the second application, we combine the classical result dealing with invariant cohomology in Lie algebroids with the Mishchenko's theorem.