Transitive Actions of Compact Groups and Topological Dimension

✍ Scribed by Karl H. Hofmann; Sidney A. Morris

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 199 KB
• Volume: 234
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2000.8543

✦ Synopsis

dedicated to helmut wielandt on the occasion of his 90th birthday

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 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 G and H denote the Lie algebras of G, respectively, H. It is shown that dim X = dim G / H . Moreover, such an X contains a space homeomorphic to 0 1 dim X ; conversely, if X contains a homeomorphic copy of a cube 0 1 ℵ , then ℵ ≤ dim X. 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.