Whitney-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 ω=(ωΔ)Δ∈K such that ωΔ is a smooth form on the Lie algebroid restriction of A to Δ, satisfying the compatibility condition concerning the restrictions of ωΔ to the faces of Δ, that is, if Δ′ is a face of Δ, the restriction of the form ωΔ to the simplex Δ′ coincides with the form ωΔ′. The set Ω∗(A;K) of all piecewise smooth forms on A

is a cochain algebra. There exists a natural morphism Ω∗(A;M)→Ω∗(A;K)

of cochain algebras given by restriction of a smooth form defined on A to a smooth form defined on the Lie algebroid restriction of A to the simplex Δ, for all simplices Δ 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 that isomorphism is induced by the restriction mapping.