Realization spaces of 4-polytopes are universal

Jürgen Richter-Gebert; Günter M. Ziegler

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arXiv · math/9510217

Realization spaces of 4-polytopes are universal

Jürgen Richter-Gebert·Günter M. Ziegler

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Abstract

Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{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.

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BibTeX RIS

Jürgen Richter-Gebert, Günter M. Ziegler. 1995-10-01. Realization spaces of 4-polytopes are universal. https://arxiv.org/abs/math/9510217

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