## Abstract

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.

Original language | English |
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Pages (from-to) | 403-412 |

Number of pages | 10 |

Journal | Bulletin of the American Mathematical Society |

Volume | 32 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1995 |

Externally published | Yes |

## Keywords

- NP-completeness
- Oriented matroids
- Polytopes
- Realization spaces
- Semialgebraic sets
- Stable equivalence
- Steinitz’s Theorem
- Universality