Combinatorial Topology - Computer Graphics and Geometric Modeling - page 328

oped. The precise nature of the invariants we are talking about cannot be defined in
a few words. We will only be scratching the surface of this topic in this chapter and
the next. In Chapter 8 we shall describe aspects of differential topology . Differential
topology builds on algebraic topology but specializes to classification problems for
the important class of differentiable manifolds. The differential structure that these
manifolds possess enables one to use constructions that are not available for general
spaces.

6.3 Simplicial Complexes

This section defines the spaces that constitute the core domain to which combinato-
rial and algebraic topology applies.

A (finite) simplicial complex K in R n is a finite collection of simplices in Definition.

R n satisfying:

(1) If s ΠK, then all faces of s belong to K.
(2) If s, t Œ K, then either s « t = f or s « t is a common face of s and t.

Definition. Let K be a simplicial complex. The underlying space of K, denoted by
ÔKÔ, is defined by

U

K =

Πs

.

s K

The dimension of K, denoted by dim K , is defined to be -1 if K is empty and the
maximum of the dimensions of the simplices of K, otherwise.

Figure 6.9 shows some examples of simplicial complexes. Note that a simplicial
complex is a set of simplices and hence not a subset of Euclidean space. Its under-
lying space is , however. In practice one is often sloppy with the terminology. In refer-
ring to the space in Figure 6.9(c), a person might very well speak of “that simplicial
complex K,” but as long as the simplices are clearly indicated, there should be no con-
fusion. In the future we may sometimes abbreviate the term “simplicial complex” to
“complex.”
Condition (1) in the definition of a simplicial complex is a technical one. Its use-
fulness will become clear later. Condition (2) is the main defining condition and basi-
cally states that one should consider a simplicial complex simply as specifying an
acceptable decomposition of a space into simplices. This can be done in many ways
however, as one can see from Figure 6.10. The two simplicial complexes K and L have
the same underlying space.

Definition. A simplicial complex L is said to be a subdivision of a simplicial complex
K if ÔKÔ=ÔLÔ and every simplex of K is a union of simplices of L.

In Figure 6.10 the simplicial complex L is a subdivision of the simplicial complex K.
A wrong way to subdivide a space into simplices is shown in Figure 6.11(a). The
set A is a set of simplices but not a simplicial complex because its two 1-simplices do