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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Euler characteristic as an obstruction to compact Lie group actions

Author: Volker Hauschild
Journal: Trans. Amer. Math. Soc. 298 (1986), 549-578
MSC: Primary 57S25; Secondary 55P62, 57R91
MathSciNet review: 860380
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Abstract: Actions of compact Lie groups on spaces $ X$ with $ {H^{\ast}}(X,{\mathbf{Q}}) \cong {\mathbf{Q}}[{x_1}, \ldots ,{x_n}]/{I_0}$, $ Q \in {I_0}$ a definite quadratic form, $ \deg {x_i} = 2$, are considered. It is shown that the existence of an effective action of a compact Lie group $ G$ on such an $ X$ implies $ \chi (X) \equiv O(\vert WG\vert)$, where $ \chi (X)$ is the Euler characteristic of $ X$ and $ \vert WG\vert$ means the order of the Weyl group of $ G$. Moreover the diverse symmetry degrees of such spaces are estimated in terms of simple cohomological data. As an application it is shown that the symmetry degree $ {N_t}(G/T)$ is equal to $ \dim G$ if $ G$ is a compact connected Lie group and $ T \subset G$ its maximal torus. Effective actions of compact connected Lie groups $ K$ on $ G/T$ with $ \dim K = \dim G$ are completely classified.

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Keywords: Topological transformation groups, cohomology theory of topological transformation groups, symmetry degree, homogeneous spaces, rational homotopy groups, spaces of $ F$-type
Article copyright: © Copyright 1986 American Mathematical Society