LECTURE 1. SYMPLECTIC FIXED POINTS AND MORSE THEORY 153of the Morse-Smale-Witten homology class [l::i m i xi] E HM* ( M, f ; Z) under the
isomorphism of Theorem 1.12. More details of the proof of Theorem 1.12 can be
found in [10, 45, 48].Corollary 1.13 (Morse inequalities). Let f : M -t JR be a Morse function and
denote by Ck the number of critical points of index k, and by bk= rankHk(M,Z)
the kth Betti number. Thenck - Ck-1 + · · · + (-l)kco;::: bk - bk-1 + · · · + (-l)kbo
for 0::::; k::::; n =dim M, and equality holds fork= n.
Proof. The weak Morse inequalities Ck ;::: bk follow fromCk= rankCMk(f);::: rankHk(CM(f), aM) =bk.The proof of the Morse inequalities in the strong form is left as an exercise. D
Exercise 1.14. Let f be a Morse function with only one critical point on each
critical level. For a E JR denote Ma = {x EM : f(x)::::; a}. Now let x be a critical
point on the level f(x) = c. Prove that, for c > 0 sufficiently small, the relative
homology of the pair (Mc+e, M c-") is given by
H (Mc+E Mc-£. Z) = { Z,
k ' ' 0,if k = ind1(x),
otherwise.Use this and the homology exact sequence for triples to deduce the Morse inequal-
ities, without resorting to Theorem 1.12. D
Exercise 1.15. A Morse function f : M _, JR is called self-indexing if f(x) =
ind / ( x) for every critical point x. In this case, prove that there is a natural iso-
morphism CMk(f) -t Hk(Mk+^112 , Mk- l /^2 ; Z) and that, under this isomorphism
the boundary operator oM corresponds to the boundary operator in the homology
exact sequence of a triple. Try to visualize this result geometrically. This can be
used to prove Theorem 1.12. D
Exercise 1.16. Identify 'JI'^2 = JR^2 / Z^2 and consider the Morse function f : 'll'^2 _,JR
given by
f(x, y) = cos(2nx) + cos(2ny).
Find the critical points and the connecting orbits (see Figure 3). Prove that f is
a Morse function with a Morse-Smale gradient flow. Compute the Morse-Smale-
Witten complex. Give an example of a gradient flow on the 2-torus which is not
Morse-Smale. D
Exercise 1.17. Consider the gradient flow on JRP^2 (thought of as the 2-disc with
opposite points on the boundary identified) which is depicted in Figure 4. Compute
the Morse-Smale-Witten complex of this example. Compute the integral homology
(and cohomology) groups of JRP^2 from the Morse-Smale-Witten complex. Find an
explicit formula of a Morse function with this gradiant flow. D