Is it true that the following two classes of subsets $\mathbb{R} ^ 3$ coincide:
- $\{f (D_ε (A)) \,\mid\, f \, \text{is an ambient isotopy, A is the union of a finite collection of closed cubes of the unit Cartesian lattice}\}$
- Compact three-dimensional submanifolds
where $D_r (A) = \bigcup\limits_{a \in A} \{x \, \mid \, d(x, a) \leq r\}$ is a closed $r$-neighborhood of the set, and $ε = \frac{1}{100}$. In other words, before union, each closed cube is increased by $\frac{1}{100}$ in all directions (thus eliminating special points of contact of the cubes along the top).
The inclusion of 1 in 2 is obvious.
P.S. It is known that all 3-manifolds are triangulable; therefore, it is not excluded that the inclusion of 2 in 1 is also true.