0

Let $G$ be a topological group acting continuously on a topological space $X$. When this action is free and proper, then $\pi:X\rightarrow X/G$ might still not be a principal fiber bundle (there might be a problem with local triviality). However, when $G$ is a Lie group, we have the following theorem (proved by Gleason for compact Lie groups and by Serre for general Lie groups):

${\bf Theorem}$: Let $G$ be a Lie group acting freely and properly on a completely regular space $X$. Then $\pi:X\rightarrow X/G$ is a principal fiber bundle.

There are some examples of Antonian and Kolmogoroff using non Lie groups visualizing this theorem. However, examples of Antonian use spaces that are not locally compact and that of Kolmogoroff uses the fact the for a space $X$ of covering dimension $n$ one might have an action of $p$-adic integers such that ${\rm dim}(X/G)>n$. See for example:

Orbit space of a free, proper G-action principal bundle

I was wondering if the following is true:

${\bf Question}$: Let $X$ be a compact Hausdorff space such that ${\rm dim}(X)=n$ and let $G$ be a compact Hausdorff group acting freely on $X$ such that ${\rm dim}(X/G)\leq n$. Then $\pi:X\rightarrow X/G$ is a principal fibre bundle.

I would be grateful for any hints, counterexamples or references.

Community
  • 1
M. Tobolski
  • 45

1 Answers1

0

Take Kolmogorov's example $G\times X\to X$ and the group $O(n)$ acting on itself via left multiplication. Now consider the product action of $G\times O(n)$ on $X\times O(n)$, $n\ge 100$.

Here is what I do not know: Suppose $X$ and $G$ are compact metrizable finite-dimensional, $G\times X\to X$ is a free action such that $dim(X/G)= dim(X) - dim(G)$. Is it true that $X\to X/G$ is a principal $G$-bundle?

Moishe Kohan
  • 111,854
  • Thank you for your answer. To be honest, I'm interested in exaclty the same situation as in your question, but for a moment I thought that it might be true for the more general case. – M. Tobolski Feb 19 '17 at 14:02
  • @M.Tobolski: If you are serious about this, you may want to email your question directly to S.Antonian (he is in Mexico and still quite active). – Moishe Kohan Feb 19 '17 at 15:25
  • If you're still interested in your question, I think I have a counterexample. Look at the paper: "On free actions of zero-dimensional compact groups" by Dranishnikov. He finds a free action of a totally disconnected $0$-dimensional metric compact group (p-adic integers for example) on a finite-dimensional compact metric space, i.e. universal Menger compactum, such that $dim(X/G)=dim(X)$. This is not locally trivial though, as n-dimensional Menger compactum is n-1-connected and n-1-locally connected. Menger compacta were described abstractly by Bestivna in his PhD thesis. – M. Tobolski Apr 29 '17 at 14:58