Whitney–Sullivan Constructions for Transitive Lie Algebroids–Smooth Case

Let \(M\) be a smooth manifold, smoothly triangulated by a simplicial complex \(K\), and \( {\cal A} \) a transitive Lie algebroid on \(M\). A piecewise smooth form on \( {\cal A} \) is a family \(\omega=(\omega_{\Delta})_{\Delta\in K}\) such that \(\omega_{\Delta}\) is a smooth form on the Lie algebroid restriction of \( {\cal A} \) to \(\Delta\), 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 \(\Omega^{\ast}( {\cal A} ;K)\) of all piecewise smooth forms on \( {\cal A} \) is a cochain algebra. There exists a natural morphism