Graphics Reference
In-Depth Information
1
2
F
£=-+£+
fc
xh
c
xh
+£+
c
e
.
Claim 2.
F and f have the same critical points.
To prove Claim 2, it suffices to prove that the only critical point of F inside
U
is
0
. But
∂
∂
F
u
∂
∂
F
∂
∂
x
∂
∂
F
∂
∂
h
=
+
x
u
h
u
i
i
i
and
∂
∂
F
(
)
<
=- - ¢
1
mx
+
2
h
0
x
∂
∂
F
(
)
≥
=- ¢ +
12
mx
2
h
1.
h
Since —x and —h only vanish at
0
, the same holds for —F and the claim is proved.
F
-1
([c-e,c+e]) is compact and contains no critical points.
Claim 3.
Claim 1 and the fact that F £ f shows that
-
1
(
[
]
)
Ã
-
1
(
[
]
)
Fc c
-+
e
,
e
f
c c
-
e
,
+
e
.
It follows that F
-1
([c-e,c-e]) is compact. The only critical point it can contain is
0
, but
this is impossible since
()
=-
()
<-
Fc
m
0
c
e
.
Claim 3 is proved.
Next, define the region
H
by
-
1
[
]
H
=
F
(
-•-
,
c
e
)
-
M
.
c
-
e
Then F
-1
([-•,c-e]) = M
c-e
»
H
. Putting all these facts together proves Theorem 8.6.3.
Definition.
The set M
c-e
»
H
is usually referred to as M
c-e
with an
attached k-handle
H
.
8.6.4. Theorem.
Let f :
M
Æ
R
be a smooth function that has only nondegenerate
critical points. Assume that
M
c
is compact for all c. Then
M
has the homotopy type
of a CW complex with one cell of dimension k for each critical point of f with
index k.
