Vol. 79, No. 1, 1978

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ISSN: 0030-8730
Combinatorial geometry and actions of compact Lie groups

Tor Skjelbred

Vol. 79 (1978), No. 1, 197–205

In this paper a theorem of combinatorial geometry will be applied to prove results about actions of compact Lie groups on manifolds.

In order to understand actions on differentiable manifolds, the weights of the tangential representations at fixed points of a maximal torus can be taken as basic data. Those weights are related by the structure of the equivariant cohomology ring of the manifold. The weights can also be considered as just a finite set of vectors or as a finite set of points in a projective space. From this point of view, theorems of combinatorial geometry can be used. Hence representation theory, equivariant cohomology theory, and combinatorial geometry can be used to understand differentiable actions. We will use the following result of combinatorial geometry which has been generalized by Sten Hansen [11]. It was conjectured by Sylvester[16] in 1893 and proved by Gallai in 1933.

Mathematical Subject Classification 2000
Primary: 57S15
Received: 1 September 1977
Published: 1 November 1978
Tor Skjelbred