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
k´
´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, Iλ
0
X U z t0uq.
Since Iλ is positive homogeneous, Iλ
0
X U radially contracts to the origin via
pIλ
0
X U q ˆ r0, 1s Ñ Iλ
0
X U, pu, tq Þ Ñ p1 ´ tq u
and Iλ
0
X U z t0u deformation retracts onto Iλ
0
X S via
pIλ
0
X U z t0uq ˆ r0, 1s Ñ Iλ
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, Iλ
0
X U z t0uq that
HqpIλ
0
X U, Iλ
0
X U z t0uq «
rq´1pIλ
H
0
X Sq.
Since Iλ|S “ I ´ λ, Iλ
0
X S “
Iλ.
We are now ready to prove Theorem 0.33.
