Realization spaces of 4-polytopes are universal

@article{RichterGebert1995RealizationSO,
  title={Realization spaces of 4-polytopes are universal},
  author={J{\"u}rgen Richter-Gebert and G{\"u}nter M. Ziegler},
  journal={Bulletin of the American Mathematical Society},
  year={1995},
  volume={32},
  pages={403-412}
}
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… 

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