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.
