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

f˚ : 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