Realization spaces of 4-polytopes are universal
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- by Jürgen Richter-Gebert and Günter M. Ziegler PDF
- Bull. Amer. Math. Soc. 32 (1995), 403-412 Request permission
Abstract:
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.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 32 (1995), 403-412
- MSC: Primary 52B11; Secondary 52B55
- DOI: https://doi.org/10.1090/S0273-0979-1995-00604-X
- MathSciNet review: 1316500