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A Class of Cellulated Spheres with Non-Polytopal Symmetries

Open Access article
 Printed: Sep 2009
  • Gábor Gévay
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We construct, for all $d\geq 4$, a cellulation of $\mathbb S^{d-1}$. We prove that these cellulations cannot be polytopal with maximal combinatorial symmetry. Such non-realizability phenomenon was first described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and, to the knowledge of the author, until now there have not been any known examples in higher dimensions. As a starting point for the construction, we introduce a new class of (Wythoffian) uniform polytopes, which we call duplexes. In proving our main result, we use some tools that we developed earlier while studying perfect polytopes. In particular, we prove perfectness of the duplexes; furthermore, we prove and make use of the perfectness of another new class of polytopes which we obtain by a variant of the so-called $E$-construction introduced by Eppstein, Kuperberg and Ziegler.
Keywords: CW sphere, polytopality, automorphism group, symmetry group, uniform polytope CW sphere, polytopality, automorphism group, symmetry group, uniform polytope
MSC Classifications: 52B11, 52B15, 52B70 show english descriptions $n$-dimensional polytopes
Symmetry properties of polytopes
Polyhedral manifolds
52B11 - $n$-dimensional polytopes
52B15 - Symmetry properties of polytopes
52B70 - Polyhedral manifolds

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