"Cohomology of some Lie algebroids and differential equations"

Lecture, II conf. GEOMETRY AND TOPOLOGY OF MANIFOLDS, Krynica, 24-29 IV 1999.
format dvi format ps publications

The fundamental examples of the Poincare duality can be found in the cohomology theories of manifolds and Lie algebras as well as in the basic cohomology for TP-foliations, Riemannian foliations, etc. The approach to this subject by means of Lie algebroids yields a common denominator for many cases, the above-mentioned and other ones. We study of transitive Lie algebroids with the trivial 1-rank adjoint bundle of isotropy Lie algebras M × R. The following differential equation on M

D = w Ů j – dj

where j Î W^m–1(M) and D Î W^m(M), m = dimM are given differential forms, is crucial for the top group of cohomology of A (and for Poincare duality). We prove that such a Lie algebroid over S¹ has the Poincare duality property for the Lie algebroid cohomology if and only if the top dimensional cohomology space is non trivial. The positive answer for arbitrary compact oriented manifold is given in [44].