The blow-down map in Lie algebroid cohomology

TL;DR

This paper develops a cohomological blow-down map for Lie algebroids, enabling the computation of cohomology for blowups and generalizing existing theorems, with applications to action Lie algebroids and de Rham cohomology.

Contribution

It introduces a Gysin sequence for Lie algebroids and extends cohomology computations to blowups of transversals of arbitrary codimension.

Findings

Computed Lie algebroid cohomology of blowups of transversals.

Extended Mazzeo-Melrose theorem to Lie algebroids.

Calculated de Rham cohomology of real projective blowups.

Abstract

We study the blow-down map in cohomology in the context of real projective blowups of Lie algebroids. Using the blow-down map in cohomology we compute the Lie algebroid cohomology of the blowup of transversals of arbitrary codimension, generalising the Mazzeo-Melrose theorem on b-cohomology. To prove the result we develop a Gysin sequence for Lie algebroids. As another example we use the developed tools to compute the Lie algebroid cohomology of the action Lie algebroid $\mathfrak{so}(3)\ltimes \mathbb{R}^3$, a result known in Poisson geometry literature. Moreover, we use similar techniques to compute the de Rham cohomology of real projective blowups.