Graphics Reference
In-Depth Information
Figure 8.22.
Deforming height function
levels into handles.
F
-1
[c-e,c+e]
p
M
c+e
e
k
H
M
c-e
8.6.3. Theorem.
Let f :
M
Æ
R
be a smooth function and let
p
be a nondegenerate
critical point for f with index k. Let c = f(
p
). Assume that f
-1
([c -e, c +e]) is compact
and has no critical point other than
p
for some e>0. Then
M
c+e
has the same homo-
topy type as
M
c-e
with a k-cell attached.
Proof.
We sketch a proof. For more details see [Miln63]. The height function for the
vertical torus is again a good example for showing what we want to do. Consider
Figure 8.22. The idea will be to deform f to a function F :
M
Æ
R
so that F is less that
f in a small neighborhood of
p
and
-
1
(
[
]
)
A
=
F
-•-
,
c
e
corresponds to M
c-e
with the horizontally lined region labeled
H
attached. By pushing
H
in along the horizontal lines to the cell e
k
one shows that
A
has the homotopy type
of
M
c-e
with a k-cell attached. But
M
c+e
can be contracted to
A
and so we will be done.
We shall now fill in some of the details.
Now since
p
is a nondegenerate critical point, it follows from Theorem 4.6.3 that
we can find local local coordinates u
i
for a neighborhood of
p
in which
p
corresponds
to the origin
0
and the function f has the form
2
2
2
2
(
)
=-
fu u
,
,...,
u
c u
-
...
-
u
+
u
+
...
+
u
12
n
1
k
k
+
1
n
in an open neighborhood
U
of the origin. The graph of f is easily analyzed in this coor-
dinate system. Figure 8.23 tries to depict the general case. To understand the picture,
try to imagine what one would see if one were to look vertically down at the critical
point
p
in Figure 8.22
Choose a sufficiently small e>0, so that
(1) f
-1
([c -e, c +e]) is compact and contains no critical point other than
p
and
(2)
U
contains the closed ball
B
of radius 2e around
p
.
Define
{
}
2
2
2
e
k
=
(
)
u
,
u
,...,
u
u
++
u
...
u
£
e
and
u
= ==
...
u
0
.
12
n
1
2
k
k
+
1
n
