14 0. MORSE THEORY AND VARIATIONAL PROBLEMS
define the Yang index of X by
ipXq inf k ě ´1 : ν˚ Hk`1pX; T q 0
(
,
taking inf H 8. Clearly, ν˚ H0pX; T q Z2 if X H, so ipXq ´1 if
and only if X H. For example,
ipSkq
k (see [133, Example 3.4]).
Proposition 0.34 ([133, Proposition 2.4]). If f : X Ñ Y is as above,
then ν˚pf˚przsqq ν˚przsq for rzs P HqpX; T q, and hence ipXq ď ipY q. In
particular, this inequality holds if X Ă Y .
Thus,
k`
´1 ď ipXq ď

´1 if there are odd continuous maps
Sk`´1
Ñ
X Ñ
Sk´´1,
so
(0.3)
γ`pXq
ď ipXq ` 1 ď
γ´pXq.
Proposition 0.35 ([98, Proposition 2.6]). If ipXq k ě 0, then
r
H kpXq
0.
Proof. We have
ν˚ HqpX; T q
#
Z2, 0 ď q ď k
0, q ě k ` 1.
We show that if rzs P HkpX; T q is such that ν˚przsq 0, then rzs 0 in
r
H kpXq. Arguing indirectly, assume that z P BkpXq, say, z Bc. Since
z P BkpX; T q, T#pzq z. Let
c1
c ` T#pcq. Then
c1
P Zk`1pX; T q since
Bc1
z ` T#pzq 2z 0 mod 2, and
ν˚prc1sq

νpc1q
νpBcq νpzq 0,
contradicting ν˚ Hk`1pX; T q 0.
Lemma 0.36. We have
CqpIλ, 0q «
r
H
q´1pIλq
@q.
Proof. Taking U u P W : }u}LppΩq ď 1
(
in (0.1) gives
CqpIλ, 0q HqpIλ
0
X U,
0
X U z t0uq.
Since is positive homogeneous,
0
X U radially contracts to the origin via
pIλ
0
X U q ˆ r0, 1s Ñ
0
X U, pu, tq Þ Ñ p1 ´ tq u
and
0
X U z t0u deformation retracts onto
0
X S via
pIλ
0
X U z t0uq ˆ r0, 1s Ñ
0
X U z t0u , pu, tq Þ Ñ p1 ´ tq u ` t u{ }u}LppΩq ,
so it follows from the exact sequence of the pair pIλ
0
X U,
0
X U z t0uq that
HqpIλ
0
X U,
0
X U z t0uq «
rq´1pIλ
H
0
X Sq.
Since Iλ|S I ´ λ,
0
X S
Iλ.
We are now ready to prove Theorem 0.33.
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