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The Euler Characteristic Of A Transitive Lie Algebroid

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.

Publication metadata

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