0.3. CRITICAL GROUPS 3
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 ;
see also Rabinowitz . The proof under the pCqc condition was given by
Cerami  and Bartolo, Benci, and Fortunato . 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
is a deformation
i.e., there is a map η P Cpr0, 1s ˆ
a deformation retraction of
piq ηp0, ¨q “ id
piiq ηpt, ¨q|Φa “ id
@t P r0, 1s,
Second deformation lemma under the pPSqc condition is due to Rothe
, Chang , and Wang . The proof under the pCqc condition can
be found in Bartsch and Li , Perera and Schechter , and in Section
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 “
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 “
q, q ě 0