On Wed, 31 Mar 2004, Richard Guy wrote:
Is it possible to turn a torus inside-out without breaking or creasing it? R.
Yes, i.e., the standard embedding of a torus in R^3 is regularly homotopic to the (relatively) inside-out embedding, just as this is the case for the sphere. The idea is, we all know from Martin Gardner that a torus T minus a disk can be turned inside out. So while keeping a small disk D of T fixed, we use that method to turn the rest of the torus inside-out. Now the only part of the torus that isn't inside-out is that small disk. By imagining D to be essentially a sphere S (while keeping a subdisk of S fixed), we can use the eversion of the sphere S to turn D inside-out as well, which completes turning the torus inside out. Dan Asimov