Mapification of n-dimensional abstract polytopes and hypertopes

Antonio Breda D'Azevedo

doi:10.26493/1855-3974.1869.9f6

Abstract

The n -dimensional abstract polytopes and hypertopes, particularly the regular ones, have gained great popularity over recent years. The main focus of research has been their symmetries and regularity. The planification of a polyhedron helps its spatial construction, yet it destroys symmetries. No “planification” of n -dimensional polytopes do exist, however it is possible to make a “mapification” of an n -dimensional polytope; in other words it is possible to construct a restrictedly-marked map representation of an abstract polytope on some surface that describes its combinatorial structures as well as all of its symmetries. There are infinitely many ways to do this, yet there is one that is more natural that describes reflections on the sides of ( n − 1) -simplices (flags or n -flags) with reflections on the sides of n -gons. The restrictedly-marked map representation of an abstract polytope is a cellular embedding of the flag graph of a polytope. We illustrate this construction with the 4 -cube, a regular 4 -polytope with automorphism group of size 384 . This paper pays a tribute to Lynne James’ last work on map representations.

Highlights

This paper stands to be a tribute to Lynne James’ last, and unfinished, work [9], where she outlines a method of representing topological categories, such as the categories of cell decompositions of n-manifolds, by other categories, for example the category of cell decompositions of oriented surfaces
For a further reading on polytopes we address the reader to the classical book by McMullen and Schulte [13]
For additional details on polytopes and related subjects we address the reader to [14]

Summary

Introduction

This paper stands to be a tribute to Lynne James’ last, and unfinished, work [9], where she outlines a method of representing topological categories, such as the categories of cell decompositions of n-manifolds, by other categories, for example the category of cell decompositions of oriented surfaces. As it turns out from the rooted Γn-slice, a restrictedly-marked map representation of an abstract polytope is a cellular embedding of the flag graph of the polytope on a connected surface without boundary. N − 1, and (R1R2)n to different, and non conjugate, elements in ∆n−1 In this way we get infinitely many distinct epimorphism and infinitely many regular restricted Γn-marked map representations of (n − 1)-hypermaps. Bold numbers and letters label the sides of this sector; the red labels signalize identifications inside the same sector, while the black ones label indentifications outside this sector Copy reflecting this sector about the central 8-gon we get the final picture of the hypercube (Figure 6) which reflects a Γ4-restrictedly regular map on an orientable surface of genus 41. The genus g of an orientable n-polytope (resp. orientable n-hypertope) can be defined to be the genus of the orientable (n − 1)-hypermap it corresponds to, which is the genus of the regular Γn-marked map representation Q without boundary

Recall that g

Ei of edges of type i given by