On the equivalence of the integrability obstructions for transitive Lie algebroids

TL;DR

This paper demonstrates that various cohomological and homotopical approaches to the integrability problem for transitive Lie algebroids are equivalent, unifying different perspectives into a single framework.

Contribution

It proves the equivalence of Mackenzie's cohomological obstruction, Crainic and Fernandes' monodromy, and Meinrenken's clutching construction for integrability.

Findings

All approaches to integrability obstructions agree and are equivalent.

The monodromy map is identified with the Mackenzie obstruction class.

The paper unifies cohomological and homotopical methods in Lie algebroid integrability.

Abstract

The integrability problem for transitive Lie algebroids can be looked at from different perspectives, revealing an interplay between cohomological methods and homotopical constructions. Mackenzie introduced a cohomological obstruction defined via sheaf-theoretic methods. On the other hand, Crainic and Fernandes used a path space approach and characterized integrability in terms of the monodromy. Recently, Meinrenken formulated the monodromy in terms of a clutching construction. We show that all of these agree. In particular, we identify the monodromy map with the Mackenzie obstruction class through the natural pairing between cohomology and homotopy.