Sullivan constructions for transitive Lie algebroids - smooth case

Abstract

Let M be a smooth manifold, smoothly triangulated by a simplicial complex K, and A a transitive Lie algebroid on M. A piecewise smooth form on A is a family of smooth forms, each one defined on the restriction of A to a simplex of K, satisfying the compatibility condition concerning the restrictions of the simplex to the faces of that simplex. The set of all piecewise smooth forms on A is a cochain algebra. One has a natural morphism from the cochain algebra of smooth forms on A to the cochain algebra of piecewise smooth forms on A given by restriction of a smooth form defined on A to a smooth form defined on the restriction of A to each simplex of K. In this paper, we prove that, for triangulated compact manifolds, the cohomology of this construction is isomorphic to the Lie algebroid cohomology of A, in which the isomorphism is induced by the restriction mapping.