Realization spaces of\\ 4-polytopes are universal

Jurgen Richter-Gebert, Guenter M. Ziegler
1995 Bulletin of the American Mathematical Society  
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
more » ... 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.
doi:10.1090/s0273-0979-1995-00604-x fatcat:ndxu3zrrunhivljsl75rmfijbu