An utterly irrelevant anecdote --- As an undergraduate, I attended a course entitled "Homology Theory" given by J. F. Adams --- a figure so august that he eventually had his own journal (in which lesser spirits like M. F. Atiyah might occasionally be granted an appendix to themselves). Halfway through the term, I felt obliged to draft a letter, to the effect that the Homology Theory wished to inform him that he had left it behind. It was signed by the entire final year class, and greeted him on the podium at the start of his next lecture. He very graciously started again from the beginning (and afterwards remarked that he intended to have the letter framed). But all in vain: for ever since, I have regarded anything to do with homology theory (or topology generally) with superstitious dread. Anyhow, the point of this tale is simply that nobody else can possibly be more hopelessly ignorant than I about homotopy theory; and any pronouncement I might be incautious enough to bring forth should treated with the gravest suspicion. Having got that out of the way, these abstract polytopes (or "polychora") are defined initially by a lattice of sets of abstract points, with an associated symmetry group [several introductory papers by Egon Schulte & co are available online.] I found it initially difficult to accustom myself to the fact that the lattice structure may be only partially realisable geometrically. In the case of Coxeter's 11-cell, although it has rank 4 --- and therefore might be expected to have associated a 3-dimensional geometrical "surface" --- the component "solid" cells would be the interiors of homeomorphs of the projective plane, which is one-sided and so doesn't have any! The way I visualise this situation is to embed the thing as a subset of a simplex in 10-space: vertices, edges and faces present no problem, but "solids" have no bounded triangulation by tetrahedra. Pretty pictures (damned if I can make sense of 'em, mind) at Carlo H. Séquin, Jaron Lanier "Hyperseeing the Regular Hendecachoron" http://www.cs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf Carlo H. Séquin, James F. Hamlin "The Regular 4-Dimensional 57-Cell" http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf Fred Lunnon On 5/28/11, Allan Wechsler <acwacw@gmail.com> wrote:
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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