Preface
The focus of this work is the study of global properties of various kinds of colorings
and maps of simplicial complexes. In addition to the usual sorts of coloring, we study
colorings determined by groups, colorings based on regular polyhedra, and continuous
colorings in finite and infinitely many colors. We are not particularly interested in either
the existence of colorings or the number of colorings, but rather in how all the colorings
fit together.
A map between two simplicial complexes X and Y is a map from the vertices of X
to the vertices of Y such that, for each r, every r-simplex of X is sent to an r-simplex of
Y. Every such map is a simplicial map, but not every simplicial map
is
of this type. An
(n+l) coloring of an n-complex is a map from X to then-simplex
ll.
n.
This is equivalent
to the usual definition. For any n-complex X, we construct an n-complex B(X) whose
n-simplices are made from the (n+l)-colorings of X. For any X and Y, the set
Hom(X,Y) of maps from X toY has the structure of ann-complex. In particular, the set
of automorphisms of X is another complex Aut(X).
The first chapter determines fundamental properties of B, Hom, Aut, and other con-
structions, such as cartesian product
# ,
join, hat, etc. For instance, Hom(X,B(Y)) is iso-
morphic to Hom(Y,B(X)). A particularly important concept is that of reflexivity. We
say that X
is
reflexive if X
is
isomorphic to B(B(X)). There is a natural map from X to
B(B(X)), but it usually
is
not an isomorphism. A simple result is that
if
X is reflexive,
then Aut(X) and Aut(B(X)) are isomorphic. An interesting calculation
is
that if R is the
6-regular triangulation of the plane, then Aut(R) is isomorphic to six disjoint copies of
R. Next, we introduce composition, and wreath product. B applied to the
composition(resp. wreath product) of X andY
is
the wreath product(resp. composition)
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