
Abstract
The main aim of this thesis is to classify the combinatorially regular
Euler incidence polytopes. The classification is completed except for
a few exceptional cases, where the subgroup structure of certain
related Coxeter groups is not sufficiently well known. The topological
structure of the objects is not considered.
The regular three dimensional (spherical) polytopes have been known
since antiquity, and the higher dimensional ones since late last
century. In recent times, various authors have attempted to abstract
the concept of regular polytopes into purely combinatorial
(nongeometric) settings. Currently, the most widely studied (but not
the only) such abstraction is Egon Schulte's regular incidence
polytopes, where a polytope is regular if and only if its
automorphism group acts transitively on its set of flags. The key
difference between this work and other work on incidence polytopes is
that instead of defining regularity in the above terms, a
combinatorial definition is used  an object being called
combinatorially regular if its local structure remains the
same all over the object (specifically, if corresponding sections are
isomorphic). It is shown that any polytope that is regular in
Schulte's sense is also regular in the combinatorial sense. This means
that combinatorial regularity is a weaker condition that the earlier
one, making the results obtained correspondingly stronger. The
Eulerian condition is a similar condition to that satisfied by the
classical geometric polytopes, and allows us to obtain an overview of
the combinatorially regular incidence polytopes. While it is true
that some authors have examined combinatorial `polytopes' that are
`less regular' than usual, a literature search reveals that objects as
`weakly' regular as those examined in the thesis have usually been
ignored. In particular, noone has attempted a classification such as
the one given.
The thesis contains a brief overview of some known results about
geometric polytopes, and then a number of combinatorial results about
the incidence polytopes are stated and proved. Some examples are given
of particular polytopes, and then a strong link is established between
the combinatorially regular incidence polytopes, and the theory of
Coxeter groups. The main theorem of the thesis is a statement of this
link, which is a classification of what might be termed ``locally
spherical'' combinatorially regular incidence polytopes, the
classification being in terms of certain subgroups of certain Coxeter
groups. Although nowhere is the classification explicitly restricted
to such `locally spherical' polytopes, this result becomes the
lynchpin of the classification theorems of the nexttolast chapter,
which attempt to describe exactly what combinatorially regular Euler
incidence polytopes exist with certain Schlafli symbols. Except for
a few particular cases, the combinatorially regular Euler incidence
polytopes are completely described.
