Graphics Reference
In-Depth Information
This leads to the next topic in this section. Given a polyhedron
X
, what is the
fewest number of cells needed in a cellular decomposition of
X
? The answer to this
question is not only important for computation purposes but also to the topological
classification of polyhedra. Exercise 6.5.5 gave a partial answer in some special cases
to the corresponding question for simplicial complexes.
Definition.
Let
c
n
be an n-cell of a CW complex C that does not belong to any higher-
dimensional cell of C. If
c
n-1
is an (n - 1)-cell of C that is contained in
c
n
but to no
other higher-dimensional cell of C, then
c
n-1
is called a
free
cell of C.
Let C be a CW complex. Let
c
n
be a top-dimensional cell in C that contains a free
(n - 1)-dimensional cell
c
n-1
. Let C¢ be the subcomplex one obtains after removing the
cells
c
n
and
c
n-1
, that is,
.
.
(
)
-
(
)
nn
n
-
1
n
-
1
CC
¢=
-
cc
-
c
-
c
.
Definition.
We shall say that C¢ is obtained from C by an
elementary collapse
from
c
n-1
through
c
n
. Conversely, we say that C is obtained from C¢ by an
elementary expan-
sion
using the cell pair (
c
n
,
c
n-1
).
See Figure 7.10. An elementary collapse through a free cell gives rise to a natural
attaching maps, so that C can be thought of as being obtained from C¢ by attaching
an n-cell.
Definition.
Let C and C¢ be CW complexes. We say that C
collapses
to C¢ and write
C Ø C¢ if there exists a sequence of CW complexes C
0
= C … C
1
… ...… C
n
= C¢ so that
C
i+1
is obtained from C
i
via an elementary collapse. In that case we also say that C¢
expands
to C.
Figure 7.10 shows a sequence of elementary collapses that collapse a disk to a
point. Figure 7.11 shows how a cell decomposition of an annulus can be collapsed to
a circle. The numbers in the 2-cells of Figure 7.11(a) indicate the order of their col-
lapse, which are then to be followed by the 1-cell collapses whose order is indicated
by the numbers in Figure 7.11(b). We end up with the circle in Figure 7.11(c).
c
0
c
0
2
2
c
1
1
c
1
c
1
2
c
0
c
0
2
c
2
c
0
c
0
1
1
1
1
c
1
c
0
c
1
c
0
c
0
c
1
3
3
3
3
3
3
collapsing
from c
1
through c
2
collapsing
from c
0
through c
1
collapsing
from c
0
through c
1
1
2
3
3
2
Figure 7.10.
Collapsing a disk to a point.
