# The Euler Characteristic Of A Transitive Lie Algebroid

## TL;DR

This paper investigates the Euler characteristic of transitive Lie algebroids, proving it vanishes unless the algebroid is the tangent bundle, and explores related cohomological properties with applications to principal bundles and Lie group cohomology.

## Contribution

It establishes a vanishing theorem for the Euler characteristic of transitive Lie algebroids and generalizes classical results like Hopf's theorem using index theory and cohomological methods.

## Key findings

- Euler characteristic vanishes unless the algebroid is the tangent bundle
- Cohomology of certain Lie algebroids are exterior algebras
- Provides a new proof for vanishing Euler characteristic of principal bundles

## Abstract

We apply the Atiyah-Singer index theorem and tensor products of elliptic complexes to the cohomology of transitive Lie algebroids. We prove that the Euler characteristic of a representation of a transitive Lie algebroid $A$ over a compact manifold $M$ vanishes unless $A=TM$, and prove a general K\"{u}nneth formula. As applications we give a short proof of a vanishing result for the Euler characteristic of a principal bundle calculated using invariant differential forms, and show that the cohomology of certain Lie algebroids are exterior algebras. The latter result can be seen as a generalization of Hopf's theorem regarding the cohomology of compact Lie groups.