Chapter 1: How To Read This Thesis

1.1 Aim of the Work

...As was stated above, the aim of the thesis is to classify the combinatorially regular Euler incidence polytopes. It can be said that this aim has been largely achieved (although in some cases the classification is not terribly explicit). Chapter 7 contains the following three theorems.

Theorem 7.1.3: If d is even, there exists a one to one correspondence between isomorphism classes of indecomposable combinatorially regular Euler d-incidence polytopes and d-CS pairs.

Theorem 7.1.4: If d is odd, there is a one to one correspondence between isomorphism classes of indecomposable combinatorially regular Euler d-incidence polytopes and finite Coxeter systems (with d generators) whose graph is a path.

Theorem 7.1.5: There is a one to one correspondence between combinatorially regular Euler incidence polytopes, and finite sequences of indecomposable combinatorially regular Euler incidence polytopes.

These theorems thrust the study of combinatorially regular Euler polytopes firmly into the study of Coxeter groups and their subgroups. The interesting thing, however, is that these results could only be derived after the bulk of the classification had been completed. Below, there is a brief outline of the thesis, which explains how this unusual state of affairs came about.

1.2 Thesis Outline

The main story starts in Chapter 3, in fact, since the second chapter is devoted to an exposition of the (no less interesting) geometric polytopes. This second chapter opens with a few technical results pertaining to the geometric polytopes, and closes with a statement of the classification of the regular ones.

Chapter 3 then begins with the definition of an incidence polytope, along with some brief notes about what is already known about these objects. Moving on to Section 3.2, we meet the Eulerian condition that was mentioned above, and discover what it means for a polytope to be `Euler'. The attack begins in earnest in Section 3.3.

In this section, a number of combinatorial results are proven. Some of these are simple, such as isomorphism, and some more deep, such as indecomposability. Still others are merely technical, included for pragmatic reasons. All contribute, in some way or other, to the final goal. In Section 3.4, the two definitions of regularity are given, and the concept of the Schlafli symbol of a combinatorially regular polytope is explained. It is shown how combinatorial regularity and the Schlafli symbol relate to the ideas and results of Section 3.3, so that all of these concepts may be used together later on. The last section of this chapter establishes a link between the incidence polytopes and the more traditional geometric ones, showing that every geometric polytope has a `combinatorial counterpart' - an Euler incidence polytope that corresponds to it in a natural way.

Chapter 4 digresses from the general study of combinatorially regular Euler incidence polytopes, and develops numerous examples of these objects. Some of the examples (such as the cubes and simplices) turn out to be isomorphic to combinatorial counterparts of geometric polytopes, whereas others (such as the halfcubes) do not. The examples are intended to aid the understanding of the classification: besides providing names for a few of the polytopes that arise, they hopefully give the reader some reasonably concrete pictures of what would otherwise be quite abstract mathematical structures.

In the fifth chapter, Coxeter groups are first encountered. A cursory overview of Coxeter group theory is given in Section 5.1.1, and Section 5.1.2 gives some additional results about those specific Coxeter groups that prove the most useful in this work. In section 5.1.3, we see developed the first explicit link between our polytopes and the Coxeter groups, and gain a taste of the power of group theoretic methods. Section 5.2 turns away, once again, from the general study of combinatorially regular Euler polytopes, and spends its pages analysing two further examples of polytopes. These are the `universal' polytope constructed from a Coxeter group, and the `quotient' polytope, constructed also from a subgroup of the Coxeter group. In particular, this section examines quotients constructed from certain particular subgroups called `sparse' subgroups. These quotients turn out to have certain nice properties, in particular that their `facets' and `vertex figures\footnot{We should note here that by their definition, the facets and vertex figures of a d-polytope are (d-1)-polytopes}' are universal (see Corollary 5.2.24). This property also enables us to calculate certain numerical results about these quotients (such as to determine exactly when they are Euler).

Section 5.3 is perhaps the key that unlocks the whole classification. Here it is shown that any incidence polytope is a quotient, and more particularly, there is a theorem (paraphrased below) about quotients by sparse subgroups.

Theorem 5.3.4: (Paraphrased) A partially ordered set is a combinatorially regular incidence polytope with universal facets and vertex figures if and only if it is a quotient by a sparse subgroup.

Why should this be the key, one might ask, when it only concerns those very special polytopes whose facets and vertex figures are universal? The answer lies in Chapter 6.

The foundation has been laid. The `Classifications' chapter builds on this foundation in a systematic manner. Sections 6.1 and 6.2 analyse in depth certain Schlafli symbols, classifying\footnot{Or almost. In actual fact, the case analysed in section 6.2 proves somewhat difficult to cover explicitly in full detail.} for all d the combinatorially regular Euler d-polytopes with those Schlafli symbols. It is shown in all cases considered, that the polytopes are of the form of the above theorem.

In Section 6.3.1, the reader is reminded that the 1- and 2-polytopes were classified in Section 3.3.1. Noting then that all 2-polytopes are universal, it is realised that all combinatorially regular 3-polytopes are covered by the theorem. An analysis of the Eulerian condition reveals then that all the indecomposable combinatorially regular Euler 3-incidence polytopes are in fact universal. It follows that the theorem also covers the combinatorially regular Euler 4-polytopes, but this time, it turns out that not all are universal.

Even so, we can use the new-found knowledge of the 4-polytopes to gain a solid grip on the 5 dimensional case, since the facets and vertex figures of a 5-polytope must be 4-polytopes. In fact, it is shown in Section 6.3.4 that if an indecomposable combinatorially regular 5-polytope has either facets or vertex figures that are not universal, then it fails to be Euler, and further, there are only a few possibilities for the Schlafli symbol of such a polytope. An inductive argument using some of the results of Section 3.4 reveals exactly what Schlafli symbols a d-polytope with d>4 can have, and they turn out to be exactly the Schlafli symbols that were covered in Sections 6.1 and 6.2. This completes the classification. Chapter 7 sums up the results of the thesis, drawing attention once again to its more interesting features, and pointing out some possible directions for future research...