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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