Realization spaces of 4polytopes are universal
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{4dimensional} 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 1995
 DOI:
 10.48550/arXiv.math/9510217
 arXiv:
 arXiv:math/9510217
 Bibcode:
 1995math.....10217R
 Keywords:

 Mathematics  Metric Geometry
 EPrint:
 10 pages