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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.

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    $\begingroup$ Can you just take the group to be the group of self-homeomorphisms? $\endgroup$ – Will Sawin Apr 11 '14 at 4:35
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    $\begingroup$ 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. $\endgroup$ – Christian Remling Apr 11 '14 at 4:53
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    $\begingroup$ @ChristianRemling: I'm thinking about the second question. Clearly we need to quotient by the stabilizer of a point. $\endgroup$ – Will Sawin Apr 11 '14 at 5:17
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    $\begingroup$ There are certainly quotients of compact groups that have the fixed point property (for instance, even-dimensional projective spaces). $\endgroup$ – Eric Wofsey Apr 11 '14 at 6:24
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    $\begingroup$ 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. $\endgroup$ – Lior Silberman Apr 11 '14 at 6:28
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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.

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