Differential Topology - Computer Graphics and Geometric Modeling - page 495

Proof. See [Miln63].

We have just seen the close connection between the nondegenerate critical points
of a real-valued function on a manifold and the manifold's topological structure. Here
is another useful variant of some theorems that show this connection.

8.6.5. Theorem. (The Weak Morse Inequalities) Let M be a closed compact smooth
n-dimensional manifold. Let c k be the number of critical points of index k for some
function f : M Æ R which has only nondegenerate critical points. Let b k be the kth
Betti number of M . Then

b £ c , and

kk

n n

k k

 Â

() = () = ()

c M 1 b 1 c .

k k

k = 0 k = 0

Proof. See [Miln63]. This is an easy consequence of Theorem 8.6.4 and properties
of the homology groups (Section 7.2.3).

We finish this section by summarizing its main results. Let M be a closed compact
connected smooth n-dimensional manifold. We know that M admits a smooth func-
tion f : M Æ R that has only nondegenerate critical points. It is not hard to show that
we may assume f( M ) = [0,n] and that k is the critical value of all critical points of f
with index k. With this assumption, we basically showed that M has (up to diffeo-
morphism) a filtration

n

DMM =ÃÃÃ = ... MM ,

0 1 n

where each M k is obtained from M k-1 by attaching as many k-handles as there are
critical points of index k. In other words,

k n - k k n - k k n - k

MM = »¥ DD »¥ DD »»¥ ... DD ,

k k - 1

where the attaching has taken place along S k-1 ¥ D n-k in each handle. Since M is con-
nected, we may assume that there is only one handle of dimension 0 and n. This handle
decomposition of a manifold is the starting point of the main classification results for
manifolds. The reader should compare this with what we know about surfaces. Each
connected closed surface can be constructed by starting with a disk and then adding
a certain number of 1-handles and finally one disk to cap it off. The nonorientable
surfaces are obtained by giving the 1-handles a “twist.”
Again let M be a closed compact connected n-dimensional manifold and consider
a smooth real-valued function

Æ [

] Ã

f : M 0 , n R

with only nondegenerate critical points and so that 0 and n are the minimum and
maximum values of f, respectively. We know that such functions f exist. This time
rather than using f to build M using handles from the bottom up (thinking of f as a
“height” function), let us build from the top down. To this end define