0.5. LINKING 9 Definition 0.22. Let A and B be disjoint nonempty subsets of W . We say that A homologically links B in dimension k if the inclusion i : A Ă W zB induces a nontrivial homomorphism i˚ : r k pAq Ñ r k pW zBq. In Examples 0.18, 0.19, and 0.20, A homologically links B in dimensions 0, dim W1 ´ 1, and dim W1, respectively. Theorem 0.23. If A homologically links B in dimension k, Φ| A ď a ă Φ| B where a is a regular value, and Φ has only a finite number of critical points in Φa and satisfies pCqc for all c ě a, then Φ has a critical point u1 with Φpu1q ą a, Ck`1pΦ,u1q ‰ 0. This follows from Theorem 0.16. Indeed, since the composition r k pAq Ñ r k pΦaq Ñ r k pW zBq induced by the inclusions A Ă Φa Ă W zB is i˚, r k pΦaq is nontrivial, and since W is contractible, it then follows from the exact sequence of the pair pW, Φaq that Hk`1pW, Φaq is nontrivial. Note that when W1 is infinite dimensional in Examples 0.19 and 0.20 the set A is contractible and therefore does not link B homotopically or homologically. Schechter and Tintarev [122] introduced yet another notion of linking according to which A links B in those examples as long as W1 or W2 is finite dimensional see also Ribarska, Tsachev, and Krastanov [113] and Schechter [120, 121]. Moreover, according to their definition of linking, if A and B are disjoint closed bounded subsets of W such that A links B and W zA is connected, then B links A. If, in addition, a :“ sup ΦpAq ď inf ΦpBq “: b and Φ is bounded on bounded sets and satisfies pCq, this then yields a pair of critical points u1 and u2 with Φpu1q ě b ě a ě Φpu2q. The following analogous result for homological linking was obtained in Perera [94], where it was shown that the second critical point also has a nontrivial critical group. We assume that Φ has only a finite number of critical points and satisfies pCq for the rest of this section. Theorem 0.24. If A homologically links B in dimension k and B is bounded, Φ| A ď a ă Φ| B where a is a regular value, and Φ is bounded from below on bounded sets, then Φ has two critical points u1 and u2 with Φpu1q ą a ą Φpu2q, Ck`1pΦ,u1q ‰ 0, CkpΦ,u2q ‰ 0. Corollary 0.25. Let W “ W1 ‘ W2, u “ u1 ` u2 be a direct sum decom- position with dim W1 “ k ă 8. If Φ ď a on u1 P W1 : }u1} ď R ( Y u “ u1 ` tv : u1 P W1, t ě 0, }u} “ R ( for some R ą 0 and v P W2 with }v} “ 1, Φ ą a on u2 P W2 : }u2} “ r for some 0 ă r ă R, where a is a regular value, and Φ is bounded from below on bounded sets, then Φ has two critical points u1 and u2 with Φpu1q ą a ą Φpu2q, Ck`1pΦ,u1q ‰ 0, CkpΦ,u2q ‰ 0.

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