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 “ ÿ uPKb a rank CqpΦ,uq. Writing βj pa, bq “ rank Hj pΦb, Φaq, we have Theorem 0.3 (Morse Inequalities). If there is only a finite number of crit- ical points in Φb, a then q ÿ j“0 p´1qq´jMj ě ÿq j“0 p´1qq´jβj , q ě 0, and 8 ÿ j“0 p´1qj Mj “ ÿ8 j“0 p´1qjβj when the series converge. Critical groups are invariant under homotopies that preserve the isolat- edness of the critical point see Rothe [116], Chang and Ghoussoub [26], and Corvellec and Hantoute [33]. Theorem 0.4. If Φt, t P r0, 1s is a family of C1-functionals on W satisfying pPSq, u is a critical point of each Φt, and there is a closed neighborhood U such that piq U contains no other critical points of Φt, piiq the map r0, 1s Ñ C1pU, 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 [14] CqpΦ, 8q “ HqpW, Φaq, 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 groups. For example, if Φ is bounded from below, CqpΦ, 8q “ δq0 G. If Φ is unbounded from below, CqpΦ, 8q “ r q´1 pΦaq where r denotes the reduced groups. 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.

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