Suppose that $X$ is a $3$-dimensional ENR continuum (a compact, connected, metric space that is a Euclidean neighborhood retract). Suppose that $X$ also satisfies the following:
- If $a, b \in X$ and $C$ is an arc in $X$, there is an isotopy $f_t$ on $X$ carrying $a$ to $b$ such that $\lbrace f_t(a) $ | $ 0 \leq t \leq 1 \rbrace$ is isotopic to $C$.
- If $C, D$ are disjoint, isotopic arcs in $X$ then there is an isotopy $f_t$ carrying $C$ onto $D$ with $f_t(C) \cap D = \varnothing$ for any $t < 1$.
- $X \setminus a$ is contractible for every $a \in X$.
Here 'arc' means the homeomorphic image of $[0,1]$.
Then is it true that $X \simeq \mathcal{S}^3$? These are potential dimension-$3$ analogues of a classification theorem for spheres of dimension greater than 4. Based on the assumptions, $X$ is a homology $3$-fold and a Cantor $3$-fold due to some standard theorems. In other words, it has the homology type of a manifold and if written as the union of two proper, closed subsets $X_1$ and $X_2$ then dim$(X_1 \cap X_2) \geq n-1$.
Also, is #2 redundant? That is probably more elementary, but I couldn't think of a rigorous proof even in the manifold case. Ofc the Bing-Borsuk Conjecture implies that $X$ is a manifold so maybe it shouldn't be much easier.