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 k pXq 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 k pXq. 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 q´1 pIλq @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, I0 λ X U radially contracts to the origin via pI0 λ X U q ˆ r0, 1s Ñ I0 λ X U, pu, tq Þ Ñ p1 ´ tq u and I0 λ X U z t0u deformation retracts onto I0 λ X S via pI0 λ X U z t0uq ˆ r0, 1s Ñ I0 λ 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 pI0 λ X U, I0 λ X U z t0uq that HqpIλ 0 X U, 0 X U z t0uq « r q´1 pIλ 0 X Sq. Since Iλ|S I ´ λ, I0 λ X S Iλ. We are now ready to prove Theorem 0.33.
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