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 dincidence polytopes and dCS pairs.
Theorem 7.1.4: If d is odd, there is a one to one
correspondence between isomorphism classes of indecomposable combinatorially regular
Euler dincidence 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 dpolytope are (d1)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
dpolytopes 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
2polytopes were classified in Section 3.3.1. Noting then that all
2polytopes are universal, it is realised that all combinatorially regular
3polytopes are covered by the theorem. An analysis of the Eulerian
condition reveals then that all the indecomposable combinatorially regular Euler
3incidence polytopes are in fact universal. It follows that the
theorem also covers the combinatorially regular Euler 4polytopes, but this time, it
turns out that not all are universal.
Even so, we can use the newfound knowledge of the 4polytopes to gain
a solid grip on the 5 dimensional case, since the facets and vertex
figures of a 5polytope must be 4polytopes. In fact, it is shown in
Section 6.3.4 that if an indecomposable combinatorially regular
5polytope 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 dpolytope 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...
