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.