Sullivan constructions for transitive Lie algebroids - smooth case

Abstract

Let $M$ be a smooth manifold, smoothly triangulated by a simplicial complex $K$, and $\cA$ a transitive Lie algebroid on $M$. The Lie algebroid restriction of $\cA$ to a simplex $\Delta$ of $K$ is denoted by $\cA^{!!}_Δ$. A piecewise smooth form of degree $p$ on $\cA$ is a family $\omega =({\omega }_{\Delta }{)}_{\Delta \in K}$ such that $ω_Δ\in Ω^{p}(\cA^{!!}_Δ;Δ)$ for each $\Delta \in K$, satisfying the compatibility condition concerning the restrictions of ${\omega }_{\Delta }$ to the faces of $\Delta$, that is, if ${\Delta }^{\prime }$ is a face of $\Delta$, the restriction of the form ${\omega }_{\Delta }$ to the simplex ${\Delta }^{\prime }$ coincides with the form ${\omega }_{{\Delta }^{\prime }}$. The set $Ω^{\ast}(\cA;K)$ of all piecewise smooth forms on $\cA$ is a cochain algebra. One has a natural morphism $$Ω^{\ast}(\cA;M)\rightarrow Ω^{\ast}(\cA;K)$$ of cochain algebras given by restriction of a smooth form defined on $\cA$ to a smooth form defined on $\cA^{!!}_Δ$, for all simplices $\Delta$ 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 $\cA$, in which the isomorphism is induced by the restriction map.

32 pages

References in corpus (1)

• Transitive Lie algebroids - categorical point of view