0.3. CRITICAL GROUPS 5

Now suppose that W is a Hilbert space pH, p¨, ¨qq and Φ P

C2pH,

Rq.

Then the Hessian A “

Φ2puq

is a self-adjoint operator on H for each u.

When u is a critical point the dimension of the negative space of A is called

the Morse index of u and is denoted by mpuq, and

m˚puq

“ mpuq`dim ker A

is called the large Morse index. We say that u is nondegenerate if A is

invertible. The Morse lemma describes the local behavior of the functional

near a nondegenerate critical point.

Lemma 0.6 (Morse Lemma). If u is a nondegenerate critical point of Φ,

then there is a local diffeomorphism ξ from a neighborhood U of u into H

with ξpuq “ 0 such that

Φpξ´1pvqq

“ Φpuq `

1

2

Av, v , v P ξpU q.

Morse lemma in

Rn

was proved by Morse [85]. Palais [92], Schwartz

[123], and Nirenberg [88] extended it to Hilbert spaces when Φ is

C3.

Proof

in the

C2

case is due to Kuiper [62] and Cambini [23].

A direct consequence of the Morse lemma is

Theorem 0.7. If u is a nondegenerate critical point of Φ, then

CqpΦ,uq “ δqmpuq G.

The handle body theorem describes the change in topology as the level

sets pass through a critical level on which there are only nondegenerate

critical points.

Theorem 0.8 (Handle Body Theorem). If c is an isolated critical value of

Φ for which there are only a finite number of nondegenerate critical points

ui, i “ 1,...,k, with Morse indices mi “ mpuiq, and Φ satisfies pPSq, then

there are an ε ą 0 and homeomorphisms ϕi from the unit disk

Dmi

in

Rmi

into H such that

Φc´ε

X

ϕipDmi

q “

Φ´1pc

´ εq X

ϕipDmi

q “

ϕipBDmi

q

and

Φc´ε

Y

Ťk

i“1

ϕipDmi

q is a deformation retract of

Φc`ε.

The references for Theorems 0.3, 0.7, and 0.8 are Morse [85], Pitcher

[106], Milnor [84], Rothe [114, 115, 117], Palais [92], Palais and Smale

[91], Smale [124], Marino and Prodi [79], Schwartz [123], Mawhin and

Willem [81], and Chang [28].

The splitting lemma generalizes the Morse lemma to degenerate critical

points. Assume that the origin is an isolated degenerate critical point of Φ

and 0 is an isolated point of the spectrum of A “

Φ2p0q.

Let N “ ker A and

write H “ N ‘ N K, u “ v ` w.

Lemma 0.9 (Splitting Lemma). There are a ball B Ă H centered at the

origin, a local homeomorphism ξ from B into H with ξp0q “ 0, and a map

η P

C1pB

X N, N

Kq

such that

Φpξpuqq “

1

2

Aw, w ` Φpv ` ηpvqq, u P B.