Graphics Reference
In-Depth Information
()
=
()
H
N
H
M
,
for
k
π-
n
1
k
k
and Corollary 7.2.3.3 proves Theorem 7.2.3.4 for these values of k.
To compute H
n-1
(N), note that
(
)
(
[
]
)
=
∂
nn
o
vv
◊◊◊
v
0
,
-
1
0
1
n
and so
(
[
]
)
Œ
()
S =
∂
n
vv
◊◊◊
v
n
ZN
.
01
n
-
1
We shall show that S is in fact a generator of Z
n-1
(N). If
n
Â
[
ˆ
]
Œ
()
z
=
a
vv
◊◊◊
v
◊◊◊
v
ZN
,
i
01
i
n
n
-
1
i
=
0
then
n
Â
()
=
(
[
ˆ
]
)
0
=
∂
z
a
∂
vv
◊◊◊
v
◊◊◊
v
n
-
1
i
n
-
1
0
1
i
n
i
=
0
n
i
-
1
n
Ê
ˆ
˜
j
j
-
1
i
Â
Â
()
[
ˆ
ˆ
]
+
()
[
ˆ
ˆ
]
=
a
1
vvvvv
◊◊◊
◊◊◊
◊◊◊
1
vvvvv
◊◊◊
◊◊◊
◊◊◊
.
Á
i
01
j
i
n
01
i
j
n
=
0
j
=
0
ji
=+
1
Let s < t. The coefficient of the oriented (n - 2)-simplex [
v
0
v
1
...
ˆ
s
...
ˆ
t
...
v
n
] is
()
+
()
-
s
t
1
1
a
1
a .
t
s
Since this coefficient has to vanish, it is easy to check that z = a
0
S. It follows that
Z
n-1
(N) =
Z
S. But B
n-1
(N) = 0 and Z
n-1
(N) has no elements of finite order, so that
()
=
()
ª
HNZN
n
Z
,
-
1
n
-
1
and the theorem is proved.
7.2.3.5. Theorem.
(1) The spheres
S
n
and
S
m
have the same homotopy type only when n = m. In
particular,
S
n
is homeomorphic to
S
m
if and only if n = m.
(2) The Euclidean space
R
n
is homeomorphic to
R
m
only when n = m.
Proof.
Part (1) follows from Theorem 7.2.3.4 and Corollary 7.2.3.2. To prove part
(2), we use the stereographic projection
n
n
p
n
:
Se
-
Æ
R
.
n
+1
