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 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, Iλ 0 X U z t0uq. Since Iλ 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, Iλ 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.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.