1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

150 D. SALAMON, FLOER HOMOLOGY

Definition 1.6. Let ( M, w) be a compact symplectic manifold. Then the minimal

Chern number of (M,w) is the integer

N = inf { k > 0 13 v : 82 ~ M, fs

2

v* c1 = k}.

If fs 2 v*c1 = 0 for every v : 82 ~ M we call N = oo the minimal Chern number.

If N =I-oo then (c1,7r2(M)) =NZ.

In the following we shall assume ( 4) and, in the case T =I- 0, normalize the

symplectic form such that fs 2 v*w E 'lJ., for every smooth map v : 82 ~ M.

1.3. The Morse-Smale-Witten complex

Let M be a compact smooth Riemannian manifold and f : M ~ IR be a Morse

function. Denote by Crit(J) = {x EM : df(x) = O} the set of critical points off.

The Morse condition asserts that the critical points are all nondegenerate. Thus

the Hessian d^2 f(x) : Tx M x TxM ~ IR is nondegenerate for every x E Crit(J).

In lo cal coordinates d^2 f(x) is given by the matrix of second partial derivatives and

the nondegeneracy condition asserts that this matrix is nonsingular.

Exercise 1. 7. Let V denote the Levi-Ci vita connection of the Riemannian metric.

Prove that the linear operator V^2 f(x) : T xM ~ T xM defined by V^2 f(x)~ =

VE. V f(x) for~ E T xM is symmetric with respect to the given Riemannian metric.

If df(x) = 0 prove that

(VE.\lf(x),TJ) = d^2 f(x)(~,TJ)

for all~' rJ E T xM. D

Consider the (negative) gradient flow

(5) u=-Vf(u)

and denote by cp^8 : M ~ M the flow of (5). The Morse condition implies that the

critical points off are hyperbolic fixed points of (5). It follows that the stable and

unstable manifolds

W^8 (x ; f) = { z EM : lim cp^8 (z) = x},

S->00

Wu(x; f) = { z EM : s~-oo Jim cp^8 (z) = x}

are smooth submanifolds of M for every critical point x of f. The Morse index of a

critical point is the number of negative eigenvalues of the Hessian (when regarded

as a linear operator V^2 f ( x)) and it agrees with the dimension of the unstable

manifold. It is denoted by

ind1(x) = v -(d^2 f(x)) =dim wu(x; f).

The gradient flow (5) is called a Morse-Smale system if, for any pair of critical

points x, y off, the stable and unstable manifolds intersect transversally. In this

case the set

M(y, x; f) = ws(x; f) n wu(y; f)