Homological properties of transitive Lie algebroids via Sullivan models

Abstract

The aim of the present work is to prove Mishchenko's conjecture. For this purpose, we have used a structure called a complex of Lie algebroids. This structure commences by xing a smooth triangulation of the base of a transitive Lie algebroid by a simplicial complex and taking the restriction of the Lie algebroid to all simplices of the triangulation. Since the Lie algebroid is transitive, the restriction of the Lie algebroid to each simplex always exists. When a complex of Lie algebroids is given, we de ne the notion of piecewise smooth form in a similar way to Whitney forms on a simplicial complex and the set of all piecewise smooth forms de ned on a complex of Lie algebroids is naturally equipped with a di erential, yielding a commutative di erential graded algebra. Its cohomology is, by de nition, the piecewise smooth cohomology of the Lie algebroid. Each smooth form de ned on the Lie algebroid gives a piecewise smooth form de ned on the corresponding complex of Lie algebroids by taking the restriction of the form to each simplex. This correspondence is a natural map from the usual algebra of the smooth forms of the Lie algebroid to the algebra of the piecewise smooth forms of the corresponding complex of Lie algebroids. Based on three crucial results, namely the triviality of a transitive Lie algebroid over a contractible smooth manifold (Mackenzie, Weinstein), the K unneth theorem for Lie algebroids (Kubarski) and the de Rham-Sullivan theorem for smooth manifolds, we show that this map is an isomorphism in cohomology.