On 5/24/11, Dan Asimov <dasimov@earthlink.net> wrote:
I don't recall if we ever discussed the 11-cell, one of only two amazing 4-dimensional "abstract regular polytopes" (ARP) built from projective planes.
Dan mentioned this in April 2007 --- but our discussion rapidly became rather technical, and took place off-list.
Also, this bizarrely symmetrical object is self-dual. I mean, who would've thought there could be such a thing with *11* faces. (The only other such thing has 55 faces -- it's 55 hemi-dodecahedra.)
Coxeter's other polytope has 57 cells.
I'd like to know more about the 11-cell as a topological object: What is its universal covering space, its homology and its homotopy?
Perhaps it's time to give more air to (some of) what was discussed then? Below, lightly edited, are notes I made at the time, sketching in particular a proof that the fundamental group of the 11-cell is trivial: I hope they're not too incomprehensibly terse! Fred Lunnon (*Def 1*) Combinatorial "road-map" representation of 11-cell --- facets indexed by "height" h = 0,...,10; addition h + [vector] increments every component modulo 11. Vertices: h + [0]; Edges: h + {[0,j], j = 1,...,5}; Faces: h + {[0,1,4], [0,1,7], [0,2,5], [0,2,7], [1,2,4]}; or h + {[1,2,5], [0,4,5], [2,4,7], [1,5,7], [4,5,7]}; Solids: S_h = h + [0,1,2,4,5,7]; The second set of faces is identical to the first: together the two lines specify the 10 faces contained in a solid. Vertices and edges contained in a face or solid are given by all possible 1- and 2-subsets. [Other incidences may be inferred from the vertex lists above; unnecessary for the task at hand, these are omitted for the present.] (*Def 2*) Symmetry group PSL(2, 11) representation by 11-permutations --- generators as products of cycles: a = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), b = (0)(10)(2, 3, 6)(1, 8, 5)(4, 7, 9). Order 660, number of conjugacy classes 8. Automorphism group PGL(2, 11) has order 1320, no 11-permutation representation. [The complete subgroup structure, relevant to thorough study of the polytope, is omitted.] Alternative "string C-group" presentation by reflections e,f,g,h: e^2 = f^2 = g^2 = h^2 = (e*f)^5 = (f*g)^3 = (g*h)^5 = (e*g)^2 = (e*h)^2 = (g*h)^2 = (e*f*g)^5 = (f*g*h)^5 = 1. [see sect. 4 of D.~Leemans, E.~Schulte "Groups of type L_2(q) acting on polytopes" (2006) http://arxiv.org/pdf/math/0606660]. (*Def 3*) Polarity (duality) between edges and faces: [0,1] -> [3,4,7], [0,2] -> [5,7,10], [0,3] -> [1,9,10], [0,4] -> [1,5,6], [0,5] -> [2,4,9]. The order-12 symmetry subgroup fixing a face fixes also its polar edge. For this polytope there is no analogous polarity between vertices and solids. (*Lem 4*) The "road map" satisfies the conditions for an abstract polytope, with incidence statistics tabulated below --- number of n-facets (across) per m-facet (down). m\n -1 0 1 2 3 4 -1 1 11 55 55 11 1 0 1 1 10 15 6 1 1 1 2 1 3 3 1 2 1 3 3 1 2 1 3 1 6 15 10 1 1 4 1 11 55 55 11 1 [Note table symmetric under half-turn, indicating self-dual polytope.] There doesn't seem any way to verify this without a great deal of inspection; frustratingly since it was originally constructed manually in a single day, entirely from the assumption that there existed a polytope of rank 4 with the given table! (*Lem 5*) The symmetry group of this polytope is PSL(2, 11). At present verification of this employs (and relies upon) the group-theoretic symbolic algebra package Magma. However it should be possible to demonstrate a set of generators corresponding to the more intuitive "string C-group" presentation reflections. Notice that the presence of 11-cycle generator "a" ensures that a road-map property valid for some height h is valid for all h modulo 11: for instance, the polarity relation extends by addition of h to every face and edge. (*Lem 6*) Any loop of edges is homotopic to a sequence of tripods, each containing the basepoint vertex. Given a sequence of vertices defining a basepointed loop L of finite length k, say L = [a, b, c, d, ..., a]. To simplify the reasoning, a vertex may be repeated consecutively; also short cases with b, c, d, ... absent are implicitly included. There are now several possibilities: Case (i): L = [a] is a singleton, incapable of further reduction. Case (ii): a = b: let L' = [a, c, ...] of length k-1. Case (iii): a = c: let L' = [a, d, ...] of length k-2. Case (iv): {a, b, c} is a face (order irrelevant); let L' = [a, c, d, ...] of length k-1. Case (v): {a, b, c} is a tripod; let L' = [a, b, c, a]*[a, c, d, ..., a] the tail of length k-2. In each case for k > 1, L' is homotopic to L and comprises a shorter tail loop L", possibly prefixed by a tripod L' = T * L" as in case (v). Now substitute L" for L and iterate. Finally k = 1, and the original loop L is reduced to a homotopically equivalent product of tripods L = T * T' * T" * ... where each factor contains the basepoint. (*Thm 7*) The fundamental group of the 11-cell is trivial. tripod [1,2,6] = [0,1,4] - [0,1,7] + [0,4,5] - [0,7,8] + [0,5,9] - [0,8,9] + [1,2,8] - [1,4,10] + [1,5,6] - [1,5,7] + [1,8,9] + [1,9,10] + [2,6,8] - [4,5,8] - [4,8,10] - [5,6,8] + [5,7,10] - [5,9,10] + [7,8,10] = 0, since every RHS term is a face; Checks out over |Z; no computer involved. QED.