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 “
Writing βj pa, bq “ rank Hj
Theorem 0.3 (Morse Inequalities). If there is only a finite number of crit-
ical points in Φa,
βj, q ě 0,
when the series converge.
Critical groups are invariant under homotopies that preserve the isolat-
edness of the critical point; see Rothe , Chang and Ghoussoub ,
and Corvellec and Hantoute .
Theorem 0.4. If Φt, t P r0, 1s is a family of
on W satisfying
pPSq, u is a critical point of each Φt, and there is a closed neighborhood U
piq U contains no other critical points of Φt,
piiq the map r0, 1s Ñ
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 
CqpΦ, 8q “ HqpW,
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
For example, if Φ is bounded from below, CqpΦ, 8q “ δq0 G. If Φ is
unbounded from below, CqpΦ, 8q “
H denotes the reduced
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.