Sullivan constructions for transitive Lie algebroids - smooth case

TL;DR

This paper establishes that for triangulated compact manifolds, the cohomology of piecewise smooth forms on a transitive Lie algebroid is isomorphic to its standard Lie algebroid cohomology, via a restriction map.

Contribution

It proves an isomorphism between piecewise smooth cohomology and Lie algebroid cohomology for triangulated compact manifolds, extending classical results.

Findings

Cohomology of piecewise smooth forms matches Lie algebroid cohomology.

Restriction map induces an isomorphism between the two cohomologies.

Validates piecewise approach for studying Lie algebroid cohomology on triangulated manifolds.

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 $\cA^{!!}_{\Delta}$. A piecewise smooth form of degree $p$ on $\cA$ is a family $\omega=(\omega_{\Delta})_{\Delta\in K}$ such that $\omega_{\Delta}\in \Omega^{p}(\cA^{!!}_{\Delta};\Delta)$ for each $\Delta\in K$, satisfying the compatibility condition concerning the restrictions of $\omega_{\Delta}$ to the faces of $\Delta$, that is, if $\Delta'$ is a face of $\Delta$, the restriction of the form $\omega_{\Delta}$ to the simplex $\Delta'$ coincides with the form $\omega_{\Delta'}$. The set $\Omega^{\ast}(\cA;K)$ of all piecewise smooth forms on $\cA$ is a cochain algebra. One has a natural morphism $$\Omega^{\ast}(\cA;M)\rightarrow \Omega^{\ast}(\cA;K)$$ of cochain algebras given by restriction of a smooth form defined on $\cA$ to a smooth form defined on $\cA^{!!}_{\Delta}$, 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.