0.7. p-LAPLACIAN 13 The eigenvalues λk are defined using the Yang index, whose definition and some properties we now recall. Yang [133] considered compact Haus- dorff spaces with fixed-point-free continuous involutions and used the Cechq homology theory, but for our purposes here it suffices to work with closed symmetric subsets of Banach spaces that do not contain the origin and sin- gular homology groups. Following [133], we first construct a special homology theory defined on the category of all pairs of closed symmetric subsets of Banach spaces that do not contain the origin and all continuous odd maps of such pairs. Let pX, Aq, A Ă X be such a pair and CpX, Aq its singular chain complex with Z2-coefficients, and denote by T# the chain map of CpX, Aq induced by the antipodal map T puq ´u. We say that a q-chain c is symmetric if T#pcq c, which holds if and only if c c1 ` T#pc1q for some q-chain c1. The symmetric q-chains form a subgroup CqpX, A T q of CqpX, Aq, and the boundary operator Bq maps CqpX, A T q into Cq´1pX, A T q, so these subgroups form a subcomplex CpX, A T q. We denote by ZqpX, A T q c P CqpX, A T q : Bqc 0 ( , BqpX, A T q Bq`1c : c P Cq`1pX, A T q ( , HqpX, A T q ZqpX, A T q{BqpX, A T q the corresponding cycles, boundaries, and homology groups. A continuous odd map f : pX, Aq Ñ pY, Bq of pairs as above induces a chain map f# : CpX, A T q Ñ CpY, B T q and hence homomorphisms : HqpX, A T q Ñ HqpY, B T q. For example, HqpSk T q # Z2, 0 ď q ď k 0, q ě k ` 1 (see [133, Example 1.8]). Let X be as above, and define homomorphisms ν : ZqpX T q Ñ Z2 inductively by νpzq # Inpcq, q 0 νpBcq, q ą 0 if z c ` T#pcq, where the index of a 0-chain c ř i ni σi is defined by Inpcq i ni. As in [133], ν is well-defined and ν BqpX T q 0, so we can define the index homomorphism ν˚ : HqpX T q Ñ Z2 by ν˚przsq νpzq. If F is a closed subset of X such that F Y T pF q X and A F X T pF q, then there is a homomorphism Δ : HqpX T q Ñ Hq´1pA T q such that ν˚pΔrzsq ν˚przsq (see [133, Proposition 2.8]). Taking F X we see that if ν˚ HkpX T q Z2, then ν˚ HqpX T q Z2 for 0 ď q ď k. We
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