Graphics Reference
In-Depth Information
c
1
1
c
2
c
2
2
1
c
1
c
1
2
3
c
2
3
(a)
(b)
(c)
Figure 7.11.
Collapsing an annulus to a circle.
c
n-1
c
n
C¢
Retracting from free cell
c
n
-
1
through cell
c
n
.
Figure 7.12.
7.2.4.5. Theorem.
Let C and C¢ be CW complexes. If C collapses to C¢, then ΩC¢Ω is
a deformation retract of ΩCΩ.
Proof.
It suffices to prove the theorem in the case where C¢ is obtained from C by
an elementary collapse from an (n - 1)-cell
c
n-1
through an n-cell
c
n
because the
general case can be proved by induction. See Figure 7.12. The result follows from the
fact that it is easy to construct a deformation retraction of
D
n
to
S
n-1
-
by “pushing
down” through
D
n
from
S
n-1
and is left as an exercise.
7.2.4.6. Corollary.
Let C and C¢ be CW complexes. If C collapses to C¢, then C and
C¢ have the same homotopy type. In fact, the inclusion map of ΩC¢Ω in ΩCΩ is a homo-
topy equivalence.
Proof.
This follows from Theorems 7.2.4.5 and 5.7.7.
Corollary 7.2.4.6 just reinforces what we have said before, namely, that homo-
logy and homotopy invariants are not good enough for detecting when spaces are
homeomorphic.
Returning to the problem of finding a minimal cell decomposition of a polyhe-
dron
X
, the idea is to start with any CW complex C with ΩCΩ=
X
and then
(1) Collapse C as much as possible to a CW complex C
1
.
(2) Pick a cell
c
1
in C
1
that does not belong to any other cell in C
1
, remove it from
C
1
, and collapse the remainder as much as possible to another CW complex C
2
.
(3) Repeat step (2) as long as there are cells
c
i
to pick.
