Combinatorial Topology - Computer Graphics and Geometric Modeling - page 333

fK L : Æ

be a simplicial map between simplicial complexes K and L. Define a map

f : K L Æ

as follows: Let x ŒÔKÔ. By Proposition 6.3.1, the point x belongs to the interior of
some unique simplex s = v 0 v 1 ··· v k of K. If the t i are the barycentric coordinates of
x with respect to s, then

k

 t ii
i

x = v ,

= 0

and we define

k

 0

() = () f x t f v .

i i

i =

Definition. The map ÔfÔ is called the map of underlying spaces induced by the sim-

plicial map f.

6.3.3. Proposition.

(1) ÔfÔ is a well-defined continuous map.
(2) ÔfÔ is a homeomorphism if and only if f is an isomorphism.
(3) If f : K Æ L and g : L Æ M are simplicial maps, then Ôg ° fÔ=ÔgÔ ° ÔfÔ.

Proof. Exercise 6.3.6.

6.4 Cutting and Pasting

We begin by discussing a slight generalization of simplicial complexes. There are two
reasons for introducing the abstract simplicial complexes defined below. One is that
simplicial complexes, sometimes called geometric complexes, play only an interme-
diate role in the study of polyhedra. It is the abstract part of their definition that one
typically exploits and not the fact that they happen to correspond to a particular sub-
division into simplices of an actual space in R n . This point will be brought home by
various constructions we carry out in this section and the next. A second reason is
that in topology one often talks about “cutting” a space or “pasting together” (or “iden-
tifying”) parts of spaces. It often helps tremendously in understanding complicated
spaces and constructions by defining them in terms of such cutting and pasting oper-
ations. The mathematical basis of these operations is the concept of a quotient space
as defined in the last chapter, but because of the special nature of what we are doing
in this chapter, we can define that concept more simply using abstract simplicial com-
plexes. Hopefully, this will also strengthen the reader's intuition about quotient spaces
in general.