Graphics Reference
In-Depth Information
Figure 7.8.
Generating a cell decomposition of a
disk.
c 1
1
c 0
c 0
c 0
c 0
2
1
1
2
(a)
(b)
c 1
c 1
1
1
c 0
c 0
c 0
c 0
c 2
1
2
1
2
1
c 1
c 1
2
2
(c)
(d)
0-cell
1-cell
a
a 1
b
c
2-cell
c 1
1-cell
b
a
b 1
c
a 3
a 2
(a)
(b)
Figure 7.9.
A compact cell decomposition for the torus.
Definition.
A subcomplex of a CW complex C is a CW complex L such that
(1) ΩLΩ is a closed subspace of ΩCΩ,
(2) ΩL n Ω=ΩLΩ«ΩC n Ω, and
(3) each cell of L is a cell of C.
Definition. A CW complex is locally finite if each of its closed cells meets only a
finite number of other cells (of any dimension). It is normal if each closed cell is a
subcomplex.
Definition. Let C and C¢ be CW complexes. A continuous map f : ΩCΩÆΩC¢Ω is called
a cellular map if f(C q ) Õ (C¢ q ) for all q.
Here are some basic facts that hold for CW complexes. Some are easy to prove
and are good exercises for the reader. The proofs of those that are not can be found
in [LunW69] or [Jäni84].
(1) Polyhedra are CW complexes. Every finite CW complex has the homotopy type
of a polyhedron and every finite regular normal CW complex can be triangulated.
We can drop the finiteness conditions here if we allow infinite simplicial complexes.
CW complexes are more general than polyhedra however. In particular, [LunW69] gives
an example of a finite three-dimensional CW complex that cannot be triangulated.
 
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