0.3. CRITICAL GROUPS 3
pviq ηp1,
Φc`εzNδpCqq
Ă
Φc´ε.
When Φ is even and C is symmetric, η may be chosen so that ηpt, ¨q is odd
for all t P r0, 1s.
First deformation lemma under the pPSqc condition is due to Palais [92];
see also Rabinowitz [108]. The proof under the pCqc condition was given by
Cerami [25] and Bartolo, Benci, and Fortunato [13]. The particular version
given here will be proved in Section 3.2.
The second deformation lemma implies that the homotopy type of sub-
level sets can change only when crossing a critical level.
Lemma 0.2 (Second Deformation Lemma). If ´8 ă a ă b ď `8 and Φ
has only a finite number of critical points at the level a, has no critical values
in pa, bq, and satisfies pCqc for all c P ra, bs X R, then
Φa
is a deformation
retract of
ΦbzKb,
i.e., there is a map η P Cpr0, 1s ˆ
pΦbzKbq, ΦbzKbq,
called
a deformation retraction of
ΦbzKb
onto
Φa,
satisfying
piq ηp0, ¨q id
ΦbzKb
,
piiq ηpt, ¨q|Φa id
Φa
@t P r0, 1s,
piiiq ηp1,
ΦbzKbq

Φa.
Second deformation lemma under the pPSqc condition is due to Rothe
[117], Chang [27], and Wang [131]. The proof under the pCqc condition can
be found in Bartsch and Li [14], Perera and Schechter [104], and in Section
3.2.
0.3. Critical Groups
In Morse theory the local behavior of Φ near an isolated critical point u
is described by the sequence of critical groups
(0.1) CqpΦ,uq
HqpΦc
X U,
Φc
X U z tuuq, q ě 0
where c Φpuq is the corresponding critical value, U is a neighborhood of u
containing no other critical points, and H denotes singular homology. They
are independent of the choice of U by the excision property.
For example, if u is a local minimizer, CqpΦ,uq δq0 G where δ is the
Kronecker delta and G is the coefficient group. A critical point u with
C1pΦ,uq 0 is called a mountain pass point.
Let ´8 ă a ă b ď `8 be regular values and assume that Φ has only
isolated critical values c1 ă c2 ă ¨ ¨ ¨ in pa, bq, with a finite number of critical
points at each level, and satisfies pPSqc for all c P ra, bs X R. Then the Morse
type numbers of Φ with respect to the interval pa, bq are defined by
Mqpa, bq
ÿ
i
rank
HqpΦai`1
,
Φai
q, q ě 0
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