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