Manifold structures on abstract regular polytopes (Q1345327)

With an abstract regular polytope can be associated its flag complex. However, this is rarely a manifold, and usually its topological structure is quite complicated. The authors here investigate different ways of associating manifolds with abstract regular polytopes. Roughly speaking, a connected \(n\)-manifold \({\mathcal M}\) is associated with an abstract regular \((n + 1)\)-polytope \({\mathcal P}\) if it admits a decomposition into compact connected submanifolds with boundary \({\mathcal M}_ F\), one for each proper face \(F\) of \({\mathcal P}\), such that \(F \leftrightarrow {\mathcal M}_ F\) is a bijection. As a motivating example \textit{H. S. M. Coxeter} and \textit{G. C. Shephard} [J. Comb. Theory, Ser. B 22, 131-138 (1977; Zbl 0338.05103)] and \textit{B. Grünbaum} [Colloq. Int. CNRS, No. 260, 191-197 (1977; Zbl 0419.05017)] separately observed that filling in the 20 toroidal facets of \(\{\{4,4\}_{(3.0)}, \{4,3\}\}\) with solid tori in an appropriate way leads to a tiling of the 3-sphere. (Some of the symmetry is lost; absent are those symmetries which interchange the ``inside'' and ``outside'' of a facet. This situation is quite general.) The authors discuss the general background to such associations, and illustrate their results with a number of examples, particularly of 4-polytopes with toroidal facets.