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
:
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.
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