Graphics Reference
In-Depth Information
Proof.
Let f :ΩKΩÆΩLΩand g : ΩLΩÆΩKΩ be continuous maps such that g
°
f
1
ΩKΩ
and f
°
g
1
ΩLΩ
. Then Fact 2 implies that
g
o
f
=
l
and
f
o
g
=
l
(
)
(
)
**
q q
HK
*
q
*
q
HL
q
q
for all q. It follows that f
*q
is an isomorphism and the theorem is proved.
7.2.3.2. Corollary.
Homotopy equivalent polyhedra have isomorphic homology
groups.
7.2.3.3. Corollary.
If a polyhedron
X
has the homotopy type of a point, then
Z
,
q=0
()
ª
Ó
˛
H
q
X
.
0,
q > 0
In particular, these are the homology groups of
D
n
.
There is one consequence of Theorem 7.2.3.1 that would be disappointing to
anyone who might have hoped to use homology groups to classify topological spaces.
They are not strong enough invariants to distinguish spaces up to homeomorphism.
For example, Corollary 7.2.3.3 shows that both a single point and the disk
D
n
have
the same homology groups but are clearly not homeomorphic. The best we could hope
for now is that they distinguish spaces up to homotopy type. Unfortunately, they fail
to do even that except in special cases. (There exist polyhedra, such as the spaces in
Example 7.2.4.7, that have isomorphic homology groups but that are
not
homotopy
equivalent.) Nevertheless, homology groups are strong enough to enable one to prove
many negative results, that is, if one can show that two spaces have nonisomorphic
homology groups, then it follows that the are
not
homeomorphic. In fact, they would
not even have the same homotopy type.
Before we state several invariance results that can be proved using homology
groups, we need to compute the homology groups for the higher-dimensional spheres.
7.2.3.4. Theorem.
If n ≥ 1, then
ZZ
Z
≈
,
if
n
=
1
and
k
=
0
Ï
Ô
Ô
¸
Ô
Ô
(
)
ª
k
S
n-1
H
,
,
if
n
≥
2
and either
k
=
0
or
k
=
n
-
1
0
if
k
π
0
or
n
-
1
.
Proof.
The case n = 1 is left as an easy exercise for the reader. Assume that n ≥ 2.
Let s =
v
0
v
1
···
v
n
be any n-simplex. Let M =·sÒ and N =∂M be the simplicial
complexes associated to the simplex and its boundary. Since
S
n-1
is homeomorphic
to ∂s =ΩNΩ, it suffices to compute H
k
(N). The definition of the simplicial homology
groups implies that
()
=
()
B
N
B
M
,
for
0
££-
k
n
2
or
k
≥
n
,
k
k
()
=
()
Z
N
Z
M
,
for all k
.
k
k
Therefore,
