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On the equivalence of the integrability obstructions for transitive Lie algebroids

Paolo Antonini, Alessio Giannotta

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.

On the equivalence of the integrability obstructions for transitive Lie algebroids

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.

Paper Structure

This paper contains 12 sections, 19 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Background on groupoids and algebroids
  3. The Mackenzie class
  4. The cohomological obstruction
  5. The Crainic--Fernandes monodromy map
  6. The Meinrenken's construction of the monodromy map
  7. Classification of transitive algebroids on the $2$--sphere.
  8. Identification of the Monodromy Homomorphisms
  9. The monodromy map as the Mackenzie class
  10. Čech to singular cohomology
  11. Ordered vs unordered coverings
  12. Darboux derivatives

Key Result

Theorem 1.1

Let $A \Rightarrow M$ be a transitive Lie algebroid. Fix a point $m \in M$ and let $\mathfrak{g}=L_m$ be the adjoint Lie algebra at $m$ with simply connected associated Lie group $\widetilde{G}$. The Crainic--Fernandes monodromy map $\partial^{\,A}: \pi_2(M,m) \to Z\widetilde{G}$ and the Meinrenken $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (37)

  • Theorem 1.1
  • Example 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Proposition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • ...and 27 more