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Does there exist a topological group which is locally homeomorphic to the Hilbert cube $[0,1]^{\mathbb N}$?

Let me note that Hilbert cube has the fixed point property and thus it is not homeomorphic to a topological group. Also, as a consequence of a recent paper by Arhangelskii and van Mill (Covering Tychonoff cubes by topological groups, Topology and its applications 2020), there is no topological group which is locally homeomorphic to $[0,1]^{\kappa}$, where $\kappa\geq\omega_1$.

A related question could be: is there a topological group structure on $\mathbb S^1\times [0,1]^{\mathbb N}$?

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  • $\begingroup$ is it locally homogeneous (i.e., is it true that any points $x,y$ have neighborhoods $V,W$ with a homeomorphism $V\to W$ mapping $x$ to $y$)? the question is the same for the $[0,1]^\omega$ and $[0,1]^\omega\times S^1$, since they're locally homeomorphic. $\endgroup$
    – YCor
    Commented Nov 8, 2021 at 14:37
  • $\begingroup$ Hilbert cube is known to be homogeneous i.e. for every $x,y\in Q=[0,1]^{\mathbb N}$ there is a homeomorphism $h:Q\to Q$ such that $h(x)=y$. $\endgroup$
    – Benjamin Vejnar
    Commented Nov 8, 2021 at 16:00

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The answer is no.

Since the Hilbert cube is compact and locally contractible, such a group would be a locally contractible locally compact group. And every locally contractible locally compact group is Lie (i.e., locally homeomorphic to $\mathbf{R}^d$ for some integer $d<\infty$).

For a reference

Szenthe, J. On the topological characterization of transitive Lie group actions. Acta Sci. Math. (Szeged) 36 (1974), 323–344. Link

Theorem 3 there: Let $G$ be a locally compact group and $H$ a closed subgroup such that the coset space $G/H$ is locally contractible. Then $G/H$ is a free [=disjoint] union of manifolds which are coset spaces of Lie groups.

(I've seen it attributed, when $H=1$ to earlier work of Gleason and Montgomery-Zippin without precise reference.)

Edit: Taras Banakh mentions in a comment that the Szenthe's proof has a gap, and that this gap is fixed independently in:

S. Antonyan, T. Dobrowolski, Locally contractible coset spaces. Forum Math. 27 (2015), no. 4, 2157–2175. DOI link

K. Hofmann, L. Kramer. Transitive actions of locally compact groups on locally contractible spaces. J. Reine Angew. Math. 702 (2015), 227–243 (+ erratum 245–246). DOI link ArXiv link

Also, user Tyrone mentions that a negative solution to the OP's question (not addressing general locally contractible locally compact groups) is the statement of Theorem 3.1 in

A. Fathi, Y. Visetti. Deformation of open embeddings of Q-manifolds. Trans. Amer. Math. Soc. 224 (1976), no. 2, 427–435 (1977). link at AMS site DOI link

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  • $\begingroup$ The same argument applies to the $[0,1]^\kappa$ case of course (the result in the paper cited in the question is much stronger than just $[0,1]^\kappa$ does not support a topological group structure though) $\endgroup$
    – Alessandro Codenotti
    Commented Nov 8, 2021 at 16:19
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    $\begingroup$ Could you quote a theorem? Every statement I have seen includes the assumption of finite-dimensionality. (As I understand it, the original question is more delicate and was an open problem for several years in the 70's. A solution appears in $\S$3 of the 1976 paper Deformation of Open Embeddings of Q-Manifolds by Fathi and Viseti.) $\endgroup$
    – Tyrone
    Commented Nov 8, 2021 at 21:50
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    $\begingroup$ @Tyrone I had quoted a theorem of the Szenthe 1974 paper. (However I'm not sure it appears in Gleason / Montgomery-Zippin, so I've modified the reference. I have also added the link to the Szenthe paper.) $\endgroup$
    – YCor
    Commented Nov 8, 2021 at 22:45
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    $\begingroup$ As far as I know there was some problem with the proof of the result of Szenthe. So, this result of Szenthe was recently reproved by Hoffmann and Kramer in (degruyter.com/document/doi/10.1515/crelle-2013-0036/html). But their proof also contained a gap, which was later fixed in (degruyter.com/document/doi/10.1515/crelle-2013-5001/html). Also the same result of Szenthe has been reproved by Antonyan and Dobrowolski in (degruyter.com/document/doi/10.1515/forum-2013-0033/html). $\endgroup$
    – Taras Banakh
    Commented Nov 12, 2021 at 6:58

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