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 : Bq c 0
(
,
BqpX, A; T q Bq`1c : c P Cq`1pX, A; T q
(
,
HqpX, A; T q ZqpX, A; T q{Bq pX, 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
ř
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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