Realization spaces of 4-polytopes are universal

Let ${P \subset \mathbb{R}^{d}}$ be a d-dimensional polytope. The realization space of P is the space of all polytopes $P \subset \mathbb{R}^{d}$ that are combinatorially equivalent to P, modulo affine transformations. We report on work by the first author, which shows that realization spaces of 4-dimensional polytopes can be "arbitrarily bad": namely, for every primary semialgebraic set V defined over ${\mathbb{Z}}$, there is a 4-polytope ${P(V)}$ whose realization space is "stably equivalent" to V. This implies that the realization space of a 4-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all 4-polytopes. The proof is constructive.

These results sharply contrast the 3-dimensional case, where realization spaces are contractible and all polytopes are realizable with integral coordinates (Steinitz's Theorem). No similar universality result was previously known in any fixed dimension.