The Euler characteristic of a transitive Lie algebroid
James Waldron
Newcastle University, Newcastle Upon Tyne, UK
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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 over a compact manifold vanishes unless , and prove a general Künneth 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.
Cite this article
James Waldron, The Euler characteristic of a transitive Lie algebroid. J. Noncommut. Geom. 17 (2023), no. 3, pp. 769–782
DOI 10.4171/JNCG/485