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Lookup NU author(s): Dr James Waldron
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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 = T M , 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.
Author(s): Waldron J
Publication type: Article
Publication status: In Press
Journal: Journal Of Noncommutative Geometry
Year: 2022
Acceptance date: 16/05/2022
ISSN (print): 1661-6952
ISSN (electronic): 1661-6960
Publisher: EMS Press
