That's a perfectly plausible explanation of how I misread Fred. I have a reflex, especially in topological contexts, to assume that when we are talking about a polyhedral complex of any sort, we are in fact talking about the surface. If it is the interior of the 11-cell that is simply-connected, I suppose that does not imply that the boundary is also (though I confess I am still a little boggled, even at that). On Fri, May 27, 2011 at 9:45 PM, Gareth McCaughan < gareth.mccaughan@pobox.com> wrote:
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
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