On Saturday 28 May 2011 02:21:20 Allan Wechsler wrote:
Doesn't the fact that the fundamental group is trivial imply that the 11-cell is homeomorphic to the 3-sphere? (This is implied by the Poincare Conjecture, now, I guess, to be called Perelman's Theorem.) Then it must be orientable.
That would be sort of weird, because the surfaces of it's 3-cells are *not*orientable. (I think they are topologically spheres with one crosscap.) But they are all nicely embedded in an orientable 3-manifold. How is this even possible? What am I missing?
My interpretation of what Fred and others wrote was: - The 11-cell is a *4-dimensional* thing. - Its fundamental group is trivial. - Its other homotopy groups may not be trivial. - In particular, what's been said so far plus the 4-dimensional version of Poincare doesn't imply that the 11-cell itself is homeomorphic to the 4-sphere. - The boundary of the 11-cell is 3-dimensional. - Its fundamental group may not be trivial. - In particular, what's been said so far plus Poincare doesn't imply that the boundary of the 11-cell is homeomorphic to the 3-sphere. ("x may not be trivial" means only "nothing I've seen said so far in this discussion obviously implies that x is trivial".) But I've forgotten most of the topology I ever knew, and never knew much about polytopes, so the above should be treated skeptically. -- g