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Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point property if every endomorphism of $X$ has a fixed-point. My question is:

Is there a example of a topological space which has the fixed-point property but is also homogeneous (that isn't the one-point space)?

The only examples of fixed-point spaces I know are compact, convex subspaces of Euclidean space which are guaranteed by the Brouwer Fixed-Point Theorem. These spaces are not homogeneous since we cannot drag a boundary point into the interior by a homeomorphism. The examples I know of homogeneous spaces are topological groups and unit spheres neither of which have the fixed-point property.

I attempted to show the properties are incompatible, but there doesn't seem to be an inherent contradiction in the two properties. I do know that topological groups do not have the fixed-point property, so those are out of the search. But other than that, my search into an example or a proof of incompatibility has been fruitless.

Any help either way is appreciated.

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  • $\begingroup$ The unit sphere in $\mathbb R^3$ doesn't have the fixed-point property? $\endgroup$
    – bof
    Jan 23, 2016 at 6:33

2 Answers 2

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The standard example is the Hilbert cube $[0,1]^\mathbb{N}$. It is homogeneous by a theorem of Keller: see Why is the Hilbert Cube homogeneous? for a discussion. It has the fixed-point property by Schauder's fixed point theorem.

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A smaller example is $\mathbb{CP}^2$. It's homogeneous since $GL_3(\mathbb{C})$ acts transitively on it and has the fixed point property by the Lefschetz fixed point theorem.

More generally, if $G$ is a Lie group and $H$ a closed subgroup of it, then $G/H$ is a homogeneous smooth manifold. This is a rich class of spaces which includes, for example, the spheres and the real and complex projective spaces as special cases. See symmetric space for more examples.

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