Abstract Polytope - The Amalgamation Problem and Universal Polytopes

The Amalgamation Problem and Universal Polytopes

The basic theory of the combinatorial structures which are now known as "abstract polytopes" (but were originally called "incidence polytopes"), was first described in Egon Schulte's doctoral dissertation, although earlier work by Branko Grünbaum, H. S. M. Coxeter and Jacques Tits laid the groundwork. Since then, research in the theory of abstract polytopes has focused mostly on regular polytopes, that is, those whose automorphism groups act transitively on the set of flags of the polytope.

An important question in the theory of abstract polytopes is the amalgamation problem. This is a series of questions such as

For given abstract polytopes K and L, are there any polytopes P whose facets are K and whose vertex figures are L ?

If so, are they all finite ?

What finite ones are there ?

For example, if K is the square, and L is the triangle, the answers to these questions are

Yes, there are polytopes P with square faces, joined three per vertex (that is, there are polytopes of type {4,3}).

Yes, they are all finite, specifically,

There is the cube, with six square faces, twelve edges and eight vertices, and the hemi-cube, with three faces, six edges and four vertices.

It is known that if the answer to the first question is 'Yes' for some regular K and L, then there is a unique polytope whose facets are K and whose vertex figures are L, called the universal polytope with these facets and vertex figures, which covers all other such polytopes. That is, suppose P is the universal polytope with facets K and vertex figures L. Then any other polytope Q with these facets and vertex figures can be written Q=P/N, where

• N is a subgroup of the automorphism group of P, and
• P/N is the collection of orbits of elements of P under the action of N, with the partial order induced by that of P.

Q=P/N is a quotient of P, and we say P covers Q.

Given this fact, the search for polytopes with particular facets and vertex figures usually goes as follows:

1. Attempt to find the applicable universal polytope
2. Attempt to classify its quotients.

These two problems are, in general, very difficult.

Returning to the example above, if K is the square, and L is the triangle, the universal polytope {K,L} is the cube (also written {4,3}). The hemicube is the quotient {4,3}/N, where N is a group of symmetries (automorphisms) of the cube with just two elements - the identity, and the symmetry that maps each corner (or edge or face) to its opposite.

If L is, instead, also a square, the universal polytope {K,L} (that is, {4,4}) is the tesselation of the Euclidean plane by squares. This tesselation has infinitely many quotients with square faces, four per vertex, some regular and some not. Except for the universal polytope itself, they all correspond to various ways to tesselate either a torus or an infinitely long cylinder with squares.