Topological group locally homeomorphic to the Hilbert cube 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 $\mathbb S^1\times [0,1]^{\mathbb N}$?
 A: 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

