4 0. MORSE THEORY AND VARIATIONAL PROBLEMS

where a “ a1 ă c1 ă a2 ă c2 ă ¨ ¨ ¨ . They are independent of the ai by the

second deformation lemma, and are related to the critical groups by

Mqpa, bq “

ÿ

uPKab

rank CqpΦ,uq.

Writing βj pa, bq “ rank Hj

pΦb, Φaq,

we have

Theorem 0.3 (Morse Inequalities). If there is only a finite number of crit-

ical points in Φa,

b

then

q

ÿ

j“0

p´1qq´j

Mj ě

ÿq

j“0

p´1qq´j

βj, q ě 0,

and

8 ÿ

j“0

p´1qj

Mj “

ÿ8

j“0

p´1qj

βj

when the series converge.

Critical groups are invariant under homotopies that preserve the isolat-

edness of the critical point; see Rothe [116], Chang and Ghoussoub [26],

and Corvellec and Hantoute [33].

Theorem 0.4. If Φt, t P r0, 1s is a family of

C1-functionals

on W satisfying

pPSq, u is a critical point of each Φt, and there is a closed neighborhood U

such that

piq U contains no other critical points of Φt,

piiq the map r0, 1s Ñ

C1pU,

Rq, t Þ Ñ Φt is continuous,

then C˚pΦt,uq are independent of t.

When the critical values are bounded from below and Φ satisfies pCq,

the global behavior of Φ can be described by the critical groups at infinity

introduced by Bartsch and Li [14]

CqpΦ, 8q “ HqpW,

Φaq,

q ě 0

where a is less than all critical values. They are independent of a by the

second deformation lemma and the homotopy invariance of the homology

groups.

For example, if Φ is bounded from below, CqpΦ, 8q “ δq0 G. If Φ is

unbounded from below, CqpΦ, 8q “

r

H

q´1pΦaq

where

r

H denotes the reduced

groups.

Theorem 0.5. If CkpΦ, 8q ‰ 0 and Φ has only a finite number of critical

points and satisfies pCq, then Φ has a critical point u with CkpΦ,uq ‰ 0.

The second deformation lemma implies that CqpΦ, 8q “ CqpΦ, 0q if

u “ 0 is the only critical point of Φ, so Φ has a nontrivial critical point

if CqpΦ, 0q ‰ CqpΦ, 8q for some q.