A FLAG VECTOR OF A 3-SPHERE THAT IS NOT THE FLAG VECTOR OF A 4-POLYTOPE

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 [Linear constraints on face numbers of polytopes. PhD Thesis, TU Berlin, Germany, 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.