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  • Print publication year: 2005
  • Online publication date: April 2013

8 - The Cohomological Obstruction

from PART TWO - THE TRANSITIVE THEORY

Summary

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.

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