Graphics Reference
In-Depth Information
D
1
¥ 0
D
1
¥ D
1
B
M
b+e
M
b-e
M
b-e
S
0
¥ D
1
A
A
Figure 8.21.
Passing a critical point adds a handle.
M
b-e
. See Figure 8.21. By a handle we mean, in this case, that we have glued a rec-
tangular strip
D
1
¥
D
1
to the boundary of
M
b-e
along
D
1
¥
S
0
. The “handle” part comes
from
0
¥
D
1
. It is not accidental that the index of the critical point b is 1, which is
also the “dimension” of the handle.
8.6.1. Theorem.
Every closed compact differentiable manifold
M
admits a smooth
real-valued function f which has only nondegenerate critical points with distinct crit-
ical values.
Proof.
See [Miln65a].
A function f for
M
of the type guaranteed by Theorem 8.6.1 clearly has at least
two critical points, namely, the two which correspond to the global minimum and
maximum of f. It is also clear that it is possible to find an f that has an arbitrary
number of nondegenerate critical points. We simply have to perturb f in a suitable
nice way. This leads to some questions. What is the minimum number of critical points
that f can have? For a fixed k, what can we say about the manifold if k is its minimum
number of critical points? A sphere clearly admits an f that has precisely two critical
points (at its north and south pole). If another manifold admits a function with only
two critical points, is it diffeomorphic to a sphere? Is there any relation between the
minimum number of critical points and algebraic invariants such as the homology
groups? These questions will be addressed in the next section. Our first order of busi-
ness is show what nondegenerate critical points imply about the local structure of a
manifold.
8.6.2. Theorem.
Let f :
M
Æ
R
be a smooth function with only nondegenerate crit-
ical points. Let a < b and assume that [a,b] does not contain any critical values of f
and that
-
1
(
[
]
)
A
=
f
ab
,
is compact. Then
A
is diffeomorphic to f
-1
(a) ¥ [0,1]. In particular,
M
a
is diffeomor-
phic to
M
b
and the inclusion map
M
a
Ã
M
b
is a homotopy equivalence.
Proof.
See [Miln63].
We may not omit the hypothesis that
A
is compact in Theorem 8.6.2.
