ABSTRACT
Let M be a compact smooth manifold, h: M —► R a suitable smooth function, and
for c e R let M c = {p € M : h(p) < c}. For c sufficiently small, M c = 0, and for c
as c varies, the topology of M c will not change except when c passes through a critical value of h\ and that when c does pass through such a critical value, the change in
points) of h. Thus the critical point structure of h will give rise to a combinatorial
model for the topology of M . For an account of this classical and powerful theory
h, so that the low-energy eigenvectors of the Laplace operators became concentrated
near the critical points of h. The object of this chapter is to give an elementary
The Morse inequalities
In Chapter 6 we defined a Dirac complex over a compact Riemannian manifold
of the various cohomology groups. If we define the Betti numbers of the Dirac complex
(S ,d) by
Pj =A\mHi (S,d), (14.1)
then Indi<•/ + d*) — £ ( —1 Yfl j - The Morse inequalities are a system of inequalities that allow one to estimate the individual Betti numbers j3j.
