Realization spaces of 4-polytopes are universal

Let P ⊂ ℝ^d be a d-dimensional polytope. The realization space of P is the space of all polytopes P′ ⊂ 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 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.