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 $\mathbb S^1\times [0,1]^{\mathbb N}$?



Topological group locally homeomorphic to the Hilbert cube
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](http://acta.bibl.u-szeged.hu/14497/1/math_036_fasc_003_004_323-344.pdf)
> 
> 
> 


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](https://doi.org/10.1515/forum-2013-0033)
> 
> 
> 



> 
> 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](https://doi.org/10.1515/crelle-2013-0036) [ArXiv link](https://arxiv.org/abs/1301.5114)
> 
> 
> 


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](https://www.ams.org/journals/tran/1976-224-02/S0002-9947-1976-0420668-4/S0002-9947-1976-0420668-4.pdf) [DOI link](https://doi.org/10.2307/1997487)
> 
> 
> 


