Take a three-dimensional torus $S^1 \times S^1 \times S^1$ and factor it by the action of permutations $S_3$ on the points (x,y,z) ~ (x,z,y) ~ e.t.c. How we can show that the quotient is homeomorphic to a solid torus?
For example if we factor $T^2$ by permutations of points, we get Möbius strip. This I can prove by considering the torus as a factor of the square. In the case of a three-dimensional torus, such a trick can not be pulled off. I was trying to figure out how to continuously connect the class of a point on the torus (which can be described by some triple of numbers) and the corresponding triple of numbers that characterizes the point on the solid torus, but for various "obvious attempts" everything collapses due to permutations.
The visual description of this homeomorphism from Wikipedia suggests that it is convenient to represent a solid torus as a "twisted" prism, where classes with pairwise different coordinates (x,y,z) will pass to the interior, to a face with 2 matching coordinates (x,x,y), and to an edge with three identical ones (x,x,x). But even with this hint, I haven't come up with anything useful yet.