On Rham cohomology of locally trivial Lie groupoids over triangulated manifolds

TL;DR

This paper proves that the Rham cohomology of locally trivial Lie groupoids over triangulated manifolds is equivalent to their piecewise Rham cohomology, which is independent of the triangulation.

Contribution

It establishes an isomorphism between Lie algebroid cohomology and piecewise smooth cohomology for such groupoids, extending cohomological invariance results.

Findings

Rham cohomology of Lie groupoids is isomorphic to piecewise Rham cohomology.

Piecewise de Rham cohomology is independent of triangulation.

The isomorphism holds for manifolds without boundary with specific transversality conditions.

Abstract

Based in the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology, it is proved that the Rham cohomology of a locally trivial Lie groupoid $G$ on a smooth manifold $M$ is isomorphic to the piecewise Rham cohomology of $G$, in which $G$ and $M$ are manifolds without boundary and $M$ is smoothly triangulated by a finite simplicial complex $K$ such that, for each simplex $\Delta$ of $K$, the inverse images of $\Delta$ by the source and target mappings of $G$ are transverses submanifolds in the ambient space $G$. As a consequence, it is shown that the piecewise de Rham cohomology of $G$ does not depend on the triangulation of the base.