On piecewise smooth cohomology of locally trivial Lie groupoids

TL;DR

This paper establishes an isomorphism between de Rham cohomology and piecewise de Rham cohomology for locally trivial Lie groupoids on combinatorial manifolds, extending previous results on Lie algebroids.

Contribution

It applies the Mishchenko-Oliveira result to locally trivial Lie groupoids, linking de Rham and piecewise de Rham cohomology for left invariant forms.

Findings

Proves isomorphism between de Rham and piecewise de Rham cohomology for certain Lie groupoids.

Extends previous cohomology isomorphism results to a broader class of groupoids.

Provides a new perspective on the cohomology of locally trivial Lie groupoids.

Abstract

Mishchenko-Oliveira proved the piecewise smooth cohomology and Lie algebroid cohomology of a Lie algebroid on a combinatorial compact manifold are isomorphic. In this paper, we describe an application of that result locally trivial Lie groupoids on combinatorial manifolds to obtain an isomorphism between de Rham cohomology and piecewise de Rham cohomology of left invariants forms.