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. \end{abstract
Article Information:
Bulletin Volume 32 Issue 4, October 1995, pages 403-412