A Flag Vector of a 3‐sphere That Is Not the Flag Vector of a 4‐polytope

Abstract

We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique 3-sphere with the parameters (f 0 , f 1 , f 2 , f 3 ; f 02 ) = (12, 40, 40, 12; 120), but this sphere is not realizable by a convex 4-polytope.The 3-sphere, which is 2-simple and 2-simplicial, was found by Werner (2009); we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: First, we show that it has no oriented matroid, and thus it is not realizable; this proof was found by computer, but can be verified by hand. The second proof is again a computer-based oriented matroid proof and shows that for exactly one of the facets this sphere does not even have a diagram based on this facet. Using the non-polytopality, we finally prove that the sphere is not even embeddable as a polytopal complex.

Small $f$-vectors of 3-spheres and of 4-polytopes

“…For all remaining candidate vectors we enumerated all compatible Eulerian lattices by Algorithm find_lattices(f), and then used the methods detailed in Brinkmann & Ziegler [11] in order to…”

“…In particular for f 0 = 11 we did not enumerate all f -vectors, but restricted ourselves to constructing polytopes. (11,22,22,11) 265 0 0 0 0 (11,23,23,11) 10 391 0 0 0 0 (11,24,24,11) 120 985 0 0 0 0 (11,25,25,11) 696 184 0 0 0 0 (11,26,26,11) 2 504 998 21 21 ≥ 1 (11,27,27,11) 6 383 318 322 322 ≥ 1 (11,28,28,11) 12 417 723 ≥ 2635 pyramids (11,29,29,11) 19 379 000 ≥ 1 (11,30,30,11) 25 121 426 ≥ 1 (11,31,31,11) 27 749 332 ≥ 1 (11,32,32,11) 26 626 961 ≥ 104 (11,33,33,11) 22 528 512 ≥ 1…”

“…The spheres with the particular f -vectors (10,32,33,11), (10,33,35,12), and (11,35,35,11) of Theorem 1.1 will be presented and discussed in Section 5.…”

“…The 2s2s property is detected by the flag vector, but not by the f -vector alone. Observation 1.2(ii) refers to all the polytopes of fatness at least 4, which in the range size(f ) ≤ 12, according to the classification of 2s2s 4-polytopes and 3-spheres of size at most 14 [11] that it is non-polytopal and that it is the only 2s2s 3-sphere with such a flag vector. As a consequence, we established that the sets of flag vectors of 4-polytopes and 3-spheres differ [11, Theorem 1.1], but did not achieve a similar statement for sets of f -vectors.…”

“…Proof. We will use a similar oriented matroid approach as in [11]. The following arguments may be verified with reference to the list of labeled facets displayed in Figure 4.…”