Graphics Reference
In-Depth Information
The groups H
q
(C) and homomorphisms f
*q
satisfy all the properties that their
simplicial analogs did. Finally, one can prove the important theorem that asserts
that if C is a finite regular normal CW complex and K is a simplicial complex with
ΩCΩ homeomorphic to ΩKΩ, then H
q
(C) is isomorphic to H
q
(K). This means that one
can obtain the homology groups of a polyhedron either from a simplicial or a cell
structure.
Definition.
If C is a finite CW complex, then let n
q
(C) denote the number of q-cells
in C and define the
Euler-Poincaré characteristic
of C, c(C), by
dim
C
Â
q
()
=
()
()
c C
1
n
C
.
q
q
=
0
The Euler-Poincaré characteristic of a CW complex is really nothing new.
7.2.4.2. Theorem.
If C is a finite regular normal CW complex, then c(C) =c(ΩCΩ).
Proof.
See [CooF67]. Note that the hypothesis implies that ΩCΩ is a polyhedron.
We know that the Euler-Poincaré characteristic and the dimension of a polyhe-
dron is a topological invariant. There is an interesting related fact.
Definition.
If f(x
0
,x
1
, . . .) is any function of the indeterminates x
i
and if C is a CW
complex, then define
()
=
(
() ()
)
fC
fn C n C
,
,... .
0
1
We shall say that f is
topologically invariant function
if f(C) = f(C¢) for all CW com-
plexes C and C¢ with homeomorphic underlying spaces.
7.2.4.3. Example.
If
•
Â
q
(
)
=-
2
(
)
=
()
=
f x
,
x
,...
x
x x
+
x
and
g x
,
x
,...
1
x
,
01
0
13
01
q
q
0
then
2
()
=
()
-
() ()
+
()
()
=
()
f C
n
C
n
C n
C
n
C
and
g C
c
C
.
0
1
3
7
7.2.4.4. Theorem.
The only topologically invariant functions f(x
0
,x
1
, . . .) on CW
complexes are those that are functions of the Euler-Poincaré characteristic and
the dimension, that is, if C and C¢ are CW complexes with dim C = dim C¢ and
c(C) =c(C¢), then f(C) = f(C¢).
Proof.
See [AgoM76].
