Topological and homological properties of the orbit space of a simple three-dimensional compact linear Lie group

TL;DR

This paper investigates the topological and homological properties of orbit spaces resulting from simple three-dimensional compact linear Lie groups, providing bounds and conditions for these spaces to be manifolds.

Contribution

It establishes an upper bound on the sum of half-dimension parts of irreducible components for representations with homological manifold quotient spaces, extending previous smooth manifold results.

Findings

Derived an upper bound for the sum of half-dimension parts in representations

Identified conditions under which the quotient space is a homological manifold

Connected most representations satisfying the bound to prior research

Abstract

The article is devoted to the question whether the orbit space of a compact linear group is a topological manifold and a homological manifold. In the paper, the case of a simple three-dimensional group is considered. An upper bound is obtained for the sum of the half-dimension integral parts of the irreducible components of a representation whose quotient space is a homological manifold, that enhances an earlier result giving the same bound if the quotient space of a representation is a smooth manifold. The most of the representations satisfying this bound are also researched before. In the proofs, standard arguments from linear algebra, theory of Lie groups and algebras and their representations are used.