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The Euler Characteristic Of A Transitive Lie Algebroid

Lookup NU author(s): Dr James Waldron

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This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).

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 = 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: Published

Journal: Journal Of Noncommutative Geometry

Year: 2023

Volume: 17

Issue: 3

Pages: 769–782

Print publication date: 11/08/2023

Online publication date: 27/05/2023

Acceptance date: 16/05/2022

Date deposited: 17/05/2024

ISSN (print): 1661-6952

ISSN (electronic): 1661-6960

Publisher: EMS Press

URL: https://doi.org/10.4171/jncg/485

DOI: 10.4171/jncg/485

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