Skip to main content Accessibility help
  • Print publication year: 2005
  • Online publication date: April 2013

8 - The Cohomological Obstruction



This chapter is concerned with the integrability problem for transitive Lie algebroids, and the single cohomologieal obstruction which gives a complete solution in that case. For general Lie algebroids, the problem of integrability has recently been completely solved; see the Notes to this chapter and the Appendix for references on the general problem.

We begin in §8.1 with a rapid summary of the most simple interesting case; most readers will know this material from accounts of geometric prequantization. In §8.2 we prove that a transitive Lie algebroid on a contractible base admits a flat connection; in terms of §5.4 this shows that every transitive Lie algebroid is locally trivial. At the same time 8.2.1 is a strong form of local integrability for transitive Lie algebroids. This result leads to a classification of transitive Lie algebroids in terms of what we call systems of transition data; these consist of a family of local Maurer–Cartan forms subject to a (nonabelian) cocycle condition twisted by a cocycle for the adjoint bundle. This classification is an exact infinitesimal analogue of the classification of principal bundles by-transition functions.

For the Lie algebroid of a locally trivial Lie groupoid, the system of transition data is obtained from a groupoid cocycle by differentiation. For an abstract transitive Lie algebroid, attempting to reverse this process leads to the eohomologieal integrability obstruction class. This is the subject of §8.3.

Related content

Powered by UNSILO