On piecewise smooth cohomology of Lie groupoids and Lie algebroids
Jose M. R Oliveira

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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Algebraic structures and combinatorial models
