The Euler Characteristic Of A Transitive Lie Algebroid

Autor:	Waldron, James
Rok vydání:	2019
Předmět:	Mathematics - Differential Geometry Mathematics - K-Theory and Homology
Druh dokumentu:	Working Paper
Popis:	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 $A=TM$, and prove a general K\"{u}nneth 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. Comment: 12 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/1908.06861 Zobrazit plný text záznamu View this record from Arxiv