5
$\begingroup$

Knaster's pseudo-arc and Hilbert cube are topologically homogeneous continua. The easier question is: do these spaces admit a topological group structure? (I am sure that the answer is negative). Thus the harder question is, do they admit a structure of a quotient of a topological group divided (left or right) by a closed (not necessarily normal) subgroup?

One could also ask extra about other non-obvious homogenous topological spaces, and also about other topological algebraic structures different from the two mentioned above, i.e. from a topological group or its quotient by a closed subgroup.

Added: I was always curious (but didn't do much about it), if the pseudo-arc can be supplied with an interesting geometric structure, even if it is made up ad hoc for the pseudo-arc.

$\endgroup$
10
  • 2
    $\begingroup$ Can you just take the group to be the group of self-homeomorphisms? $\endgroup$
    – Will Sawin
    Apr 11, 2014 at 4:35
  • 2
    $\begingroup$ The HC indeed admits no group structure because every continuous self map has a fixed point (use Brouwer's fixed point theorem on $n$-dimensional projections, then pass to a limit on a subsequence), but of course $x\mapsto ax$ for $a\not= e$ doesn't. $\endgroup$ Apr 11, 2014 at 4:53
  • 1
    $\begingroup$ @ChristianRemling: I'm thinking about the second question. Clearly we need to quotient by the stabilizer of a point. $\endgroup$
    – Will Sawin
    Apr 11, 2014 at 5:17
  • 2
    $\begingroup$ There are certainly quotients of compact groups that have the fixed point property (for instance, even-dimensional projective spaces). $\endgroup$ Apr 11, 2014 at 6:24
  • 2
    $\begingroup$ The quotient $G/H$ has the fixed point property exactly when the conjugates of $H$ cover $G$. This can't happen for finite groups (conjugate by $G/N_G(H)$ and count the elements) but can happen for infinite groups. In Eric's example $G$ is a compact Lie group and $H$ is any subgroup containing a maximal torus. $\endgroup$ Apr 11, 2014 at 6:28

1 Answer 1

5
$\begingroup$

As it was mentioned in the comments, the pseudo-arc and the Hilbert cube have the fixed point property so they cannot be homeomorphic to a topological group.

On the other hand it was proved by G.S. Ungar in "On all kinds of homogeneous spaces" (TAMS, 1975), that any homogeneous compact metric space is homeomorphic to a coset space. In particular this is true for both the pseudo-arc and the Hilbert cube.

The fact that the pseudo-arc is a coset space was first proved by T.S. Wu in "Each homogeneous nondegenerate chainable continuum is a coset space" (PAMS, 1961).

I don´t know who proved first that the Hilbert cube is a coset space, but it also follows from a theorem of L.F. Ford in "Homeomorphism groups and coset spaces" (TAMS, 1954), namely that any homogeneous strongly locally homogeneous Tychonoff space is a coset space.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.