Deformation Cohomology of Lie Algebroids and Morita Equivalence

TL;DR

This paper proves that the deformation cohomology of a Lie algebroid remains invariant under pull-backs along fibrations with sufficiently connected fibers, extending understanding of cohomological invariance in Lie algebroid theory.

Contribution

It establishes a new invariance property of deformation cohomology for Lie algebroids under specific pull-back operations involving fibrations.

Findings

Deformation cohomology is preserved up to degree k under pull-back along fibrations with homologically k-connected fibers.

Provides a new tool for comparing Lie algebroids via their deformation cohomology.

Enhances understanding of Morita equivalence in the context of Lie algebroids.

Abstract

Let $A \Rightarrow M$ be a Lie algebroid. In this short note, we prove that a pull-back of $A$ along a fibration with homologically $k$-connected fibers, shares the same deformation cohomology of $A$ up to degree $k$.