Homological properties of transitive Lie algebroids via Sullivan models

Resumo

D. Sullivan considered a new model for the underlying cochain complex of classical cohomologies with rational coefficients for arbitrary simplicial spaces which gives an isomorphism with classical rational cohomologies. This new model is determined by the Rham complex of all rational polynomial forms defined on the simplicial complex triangulating the space. Other cell-like constructions of cochain complexes which induce isomorphisms in cohomology with classical cohomologies had been already presented by H. Whitney. Recent ideas developed by K. Mackenzie and J. Kubarski concerning Lie algebroids are applied to a generalization of a cell-like construction for transitive Lie algebroids over combinatorial manifolds. Namely, given a compact smooth manifold M, smoothly triangulated by a simplicial complex K, and a transitive Lie algebroid A on M, we define a piecewise smooth form on A to be a family w of differential forms such that, for each simplex a of K, the family w in a is a smooth form defined on the restriction of the Lie algebroid A to the simplex a, satisfying the compatibility condition under restrictions of the form w in a to all faces of the simplex a, that is, if b is a face of a, then the form w in b is the restriction of the form w in a to the face b. The set Wps(A;K) of all piecewise smooth forms defined on A is a cochain algebra. We define a map from the cochain algebra W(A;M) of all smooth forms on A to the cochain algebra Wps(A;K) which assigns, to each smooth form w on A, the unique piecewise smooth form on A defined by the restriction of w to each simplex a of K. In this thesis, we prove that, for compact combinatorial manifolds, the cohomology of this construction is isomorphic to the Lie algebroid cohomology of A. We apply this isomorphism in piecewise invariant cohomology of Lie algebroids and piecewise de Rham cohomology of locally trivial Lie groupoids.