Transitive Actions of Compact Groups and Topological Dimension

Abstract

There are many dimension functions defined on arbitrary topological spaces taking either a finite value or the value infinity. This paper defines a cardinal valued dimension function, dim. The Lie algebra L(G) of a compact group G is a weakly complete topological vector space. Quotient spaces of weakly complete spaces are weakly complete; the dimension of a weakly complete vector space is the linear dimension of its dual. Assume that a compact group G acts transitively on a given space X and that H is the isotropy group of the action at an arbitrary point; let L(G) and L(H) denote the Lie algebras of G, respectively, H. It is shown that dimX=dimL(G)/L(H). Moreover, such an X contains a space homeomorphic to [0,1]dimX; conversely, if X contains a homeomorphic copy of a cube [0,1]ℵ, then ℵ≤dimX. En route one establishes a good deal of information on the quotient spaces G/H; such information is of independent interest. Finally, these results are generalized to quotient spaces of locally compact groups. A generalization of a theorem of Iwasawa is instrumental; it is of independent interest as well.