Daniel Pellicer: Abstract regular and chiral polytopes && Topological structure and amalgamation of abstract polytopes

Abstract regular polytopes were introduce as a consequence of successive generalizations of the concept of regular polytopes. This naturally motivated the concept of abstract polytope. Regular and chiral abstract polytopes are highly symmetric polytopes and have attracted attention in recent years. We shall provide the necessary definitions as well as some (combinatorial and algebraic) properties, results and examples.

 

Topological structure and amalgamation of abstract polytopes 

 

Some classifications of abstract polytopes according to their topological local structure have been done. The efforts made in this direction include attacking the problem of determining the existence of a universal polytope with particular local conditions. We shall discuss locally spherical, locally toroidal and locally projective polytopes. Also related to these topics are the problem of amalgamating a polytope with facets and vertex-figures isomorphic to given polytopes P and Q respectibly, and the problem of determining the existence of a regular polytope with facets isomorphic to a given polytope P and last entry of the Scläfli type equal to a given number n.