Generalization of the Sullivan construction for Transitive Lie Algebroids

TL;DR

This paper extends Sullivan's rational cohomology model to transitive Lie algebroids, utilizing properties like triviality over contractible manifolds and inverse images to develop a cochain complex for simplicial spaces.

Contribution

It generalizes Sullivan's model for classical cohomologies to the setting of transitive Lie algebroids, incorporating key ideas from MacKenzie and Kubarski.

Findings

Constructed a cochain complex for transitive Lie algebroids on simplicial spaces.

Demonstrated the model's consistency with classical rational cohomology.

Extended the applicability of Sullivan's approach to a broader geometric context.

Abstract

D.Sullivan (1977) (see also the book by H.Whitney "Geometric Integration Theory",1957) considered a new model for underlying cochain complex for classical cohomologies with rational coefficients for arbitrary simplicial spaces that gives the isomorphism with classical rational cohomologies. We apply the key ideas developed by K.MacKenzie (2005) and J.Kubarski (1991) to generalization of the D.Sullivan model for transitive Lie algebroids. The generalization is based on the existence of the inverse image of the transitive Lie algebroids and on the property of transitive Lie algebroids being trivial over contractible manifolds. Using these properties one can construct an underlying cochain complex of differential forms on simplicial space.