Algebraic Topology - Computer Graphics and Geometric Modeling - page 391

The space X is called the underlying space of the CW complex C and denoted by
ΩCΩ. If the subspace structure on X defined by C is clear from the context, one often
uses the phrase “the CW complex X ” to refer to C. We define C n , called the n-skeleton
of C, to be the CW complex with underlying space X n

and sequence of closed

subspaces

- 1 0 1 n - 1 n n

XXX ÃÃÃ . . . X XX ÃÃÃ ....

The subspace X n is usually also called the n-skeleton of C. A CW complex C is finite
if it has only finitely many cells. It has dimension n if it has at least one n-cell but no
m-cells for m > n (equivalently, ΩC n Ω=ΩC n+1 Ω=. . .). If C has dimension n for some
n (denoted by dim C = n), then it is said to be finite dimensional , otherwise, it is said
to be infinite dimensional . If the characteristic maps are all homeomorphisms, then
the CW complex is said to be regular , otherwise, it is irregular .

CW complexes were first defined by J.H.C. Whitehead in 1949. The “C” stands for
“closure finite” and the “W” for “weak topology.” The term closure finite is the tech-
nical term for the property that every closed n-cell of a CW complex C meets only a
finite number of open cells in ΩC n-1 Ω. For a proof that this property is satisfied by a
CW complex as defined above see [LunW69]. Whitehead's definition of a CW complex
actually differed slightly from the one given here. He started with abstract cell decom-
positions for which that property had to be stipulated. One can easily show that con-
dition (4) of the definition is trivially satisfied for finite CW complexes and can be
omitted if one restricts oneself to such complexes.
A CW complex defines a cell decomposition for a space. One normally thinks of
a CW complex as a space where one start with some 0-cells (points), then attaches
some 1-cells, then some 2-cells, and so on. Cells are to CW complexes what sim-
plices are to simplicial complexes. (Note that the initial set of points can also be
thought of in terms of having attached some 0-cells to an initially empty space.)
Clearly, every simplicial complex is a regular CW complex because it defines an
obvious sequence of cells and skeletons. The main difference between the cell decom-
position induced on a space X by a regular CW complex and a triangulation of X
is that cells have a flexible number of faces and are potentially “curved” from the
start.
Figure 7.8 shows the inductive aspect of the definition of a regular CW complex
by showing the steps that represent a disk as a regular CW complex. We started with
two points and then attached two 1-cells and one 2-cell. It is easy to construct a regular
cell decomposition for a torus by dividing a rectangle into four equal subrectangles
and identifying the boundary pieces appropriately. This structure will have four 0-
cells, eight 1-cells, and four 2-cells and is quite an improvement, in terms of numbers
of cells, over the standard triangulation of the torus shown in Figure 7.5. In fact,
we can even do better. Figure 7.9 shows a CW complex whose underlying space
is the torus and which consists of four cells - one 0-cell a , two 1-cells b and c , and
one 2-cell. The boundary of the 2-cell in Figure 7.9(a) is mapped onto the 1-skeleton
a » b » c in Figure 7.9(b) by mapping the edges b 1 and c 1 to b and c , respectively,
using the orientation indicated by the arrows. This will send the vertices a and a i to
a . Note that if we were to cut along the circles b and c in the torus we would unfold
it to a rectangle.