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˚ :

rk

H pAq Ñ

rk

H 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

rk

H pAq Ñ

rk

H

pΦaq

Ñ

rk

H pW zBq induced by the inclusions A Ă

Φa

Ă W zB is i˚,

r

H

kpΦaq

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.