Lecture 3.
3.1. Classical Morse theory
In this lecture we begin the proof of Theorem 1.1. We start by discussing t he casewhen Xis compact, and Ji = h = 1 (cf. Exercise 1.5.2).
We have x(f 1 ·h) = x(l) = x(X), the Euler ch aracteristic of X. Also, Ch(f 1 ) =
(-l)Ch(f1) = Ch(h) =Ax, the zero section of T X (with multiplicity one). We
need to show that Ax n Ax= x(X).
There are many way of doing this. We shall go quickly through the Morse
theoretic proof, mostly to prepare for a discussion of stratified Morse theory in the
next section.We choose a smooth function g: X--> IR. Let {dg} CT* X be the graph of dg.
Then {dg} is isotopic to Ax in T* X, and we have Ax n Ax =Ax n {dg}. The
intersection points of Ax and { dg} correspond to the critical points of g. Recall
that a critical point p E X is called Morse if the Hessian matrix H p(g) of second
partials at p is invertible.Exercise 3.1.1. A crit ical point p of g is Morse if and only if { dg} meets Ax
transversely at (p, 0) ET* X.Let p be a Morse critical point. Recall that the index .A(p) of p is defined as
the number of negative eigenvalues of Hp(g).Exercise 3.1.2. Show that t he sign of t he intersection of Ax with { dg} at (p, 0)
is equal to (-1)>-(p).In fact, t here is a symplectic interpretation of the index itself. We won't need
it, but suggest it as an exercise.Exercise 3.1.3. Let (V^2 n,w) be a symplectic vector space. Denote by E the
set of all ordered triples (L 1 , L2, L 3) of pairwise transverse Lagrangian subspaces
of V. Show that E has n + 1 connected components, and that we can label
them by the set {O, ... ,n}, so that .A(p) is the label corresponding to the triple
(T(p,o)Ax, T(p,o){dg}, T (p ,o)A{p}) of Lagrangians in T(p,o)T * X.
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