The Euler characteristic of a transitive Lie algebroid

Journal of noncommutative geometry

article

James Waldron¹ 

[1] Newcastle University

关键词: Differential geometry;    Lie algebroids;    index theory;    H -spaces;   

DOI  :  10.4171/jncg/485

学科分类：神经科学

来源: European Mathematical Society

【 摘 要 】

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 AAA over a compact manifold MMM vanishes unless A=TMA=TMA=TM, 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.

【 授权许可】

CC BY