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.
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