Topological and Homological Properties of the Orbit Space of a Simple Three-Dimensional Compact Linear Lie Group

Abstract

The question of whether an orbit space of a compact linear group is a topological manifold and a homology manifold is studied. The case of a simple three-dimensional group is considered. An upper bound is obtained for the sum of integral parts of the halved dimensions of irreducible components for a representation whose quotient is a homology manifold. This strengthens a similar result obtained previously, which gave such a bound in the case where the quotient of the representation is a smooth manifold. Most representations for which the obtained estimate holds have also been considered previously. The argument uses standard considerations of linear algebra and the theory of Lie groups and algebras and their representations.