Open-problem statements, with their sources attached.

Hilbert–Smith conjecture*: every locally compact topological group acting continuously and faithfully on a connected finite-dimensional topological manifold is a Lie group.

The four dimensional case of the smooth version of the conjecture is still open. See [Wang2017].

Equivalent p-adic formulation: the p-adic integers ℤ_[p] cannot act continuously and faithfully on any connected finite-dimensional topological manifold. By the Gleason–Yamabe theorem, this is equivalent to hilbert_smith_conjecture.

It is conjectured that the only values of $n > 4$ for which the smooth version of the conjecture holds are $n = 5, 6, 12, 56, 61$. See Conjecture 1.17 in [Wang2017].

The Bing-Borsuk Conjecture: every $n$-dimensional homogeneous absolute neighborhood retract is a topological $n$-manifold. A topological space $X$ is an $n$-dimensional manifold when T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X). The hypothesis [MetrizableSpace X] implies T2Space X so this does not appear in the conclusion.