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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