Regular Incidence-Complexes and Dimensionally Unbounded Sequences of Such, I

Manifold structures on abstract regular polytopes

“…Hence (8) implies H~ (.It') ---Z g. But then the Heegaard genus of ~¢/cannot be smaller than g because it bounds the rank of HI (J¢). In order to prove (8) we make a suitable choice of 2g generators for H1 (S) and prove that 2 maps g of them to 0. The relation g = 4g 0 +f2 -3 indicates a way how these g generators can be found.…”

“…We also briefly mention Danzer's polytopes 2 ~ for a given n-polytope ~ ( [8]). Assume that ~ is a lattice with vertex set V = { 1 ..... v } and that the facets of Jf are (n -1)-simplices; then ~" can be regarded as a simplicial (n -l)-complex with vertex set V. For each /-face F of W with vertex set VF and each j e V define …”

Chiral extensions of chiral polytopes

“…We denote by {4, 4}ū the quotient of the regular tessellation {4, 4} by the subgroup Tū of T generated by the translational symmetries with respect to the linearly independent vectorsū andūR, where R stands for the rotation by π/2. The chiral polyhedra {4, 4} (2,1) and {4, 4} (4,1) are shown in Figure 1.…”

Developments and open problems on chiral polytopes

“…There are regular extensions of each regular polytope K. Furthermore, if K is finite, it admits finite regular extensions (see for example [20] and [53]). The Schläfli type of any equivelar extension of an equivelar polytope K will coincide on the first d − 1 entries with the Schläfli type of K. It was determined in [48] that every regular polytope admits regular extensions where the last entry of the Schlälfi type is an arbitrary even number p d .…”