Ghoussoub , Min-max critical points 51

Proof , Assume

Iné^iK . nFnAJ й Indçi^, D) - lnd(^(FnB) - m.

Since F n С = 0, we can use property (13) of the index to find an invariant neighbourhood U of K^nFnA^ that is disjoint from С and such that Indç(C7) = Ind^(Ä^nFn^^). Similarly, find an invariant neighbourhood VoîFnB that is disjoint from С and such that Indö(F) = Ind^(Fn5).

For any A in #", we have by property (R3) of the relative index

Ind^ { A , D) й Ind^iiA\U\F), D) + Ind^{K, nFnAJ-\- Ind^(BnF).

It follows that IndG(.4 \U\V) ^ m, hence A \U\VeI^(D) and {A \U\V)nF+0, for all v4 in J^. In other words, if we set F' = F \ f/ \ F, then F' verifies (Fl ) and (F2) with respect to the class #". It follows that F' пК^пА^ Ф 0 which is a contradiction.

Remark (9). (a) If iî = 0 and ^ = I„ for some n^m, then (F4) reduces to the condition c„ = c^. We then recover the classical estimate Ind(j(Ä^^) ^n — m-\-i by just taking F ={ф^с„}.

( b ) We will have the same estimate as in (a) if we take D = 0 (the regular index), С = B, sup ф(В) < с and J^ = rf. A sHght refinement of the above proof will actually give the following formally stronger statement:

Assume sup ф{В) < c^ = c„for n^m where c^ = с(ф,1^) and

if = {A\ A compact invariant containing В and Ind(j(^) ^ i), Then for any compact invariant set К with 1па^{К) -^n — mwe have K^^\K Ф 0.

More generally (i.e. if we do not suppose sup 0(5) < c) we have the following

Corollary (12). Let G, ф, ^ and В be as in Proposition (1). Assume В invariant and suppose that for some e > 0 and m^\ we have с{ф,^) = с{ф, E^^) where

B^ = Вп{ф^с-е}.

Then we have

Ind^iK , ) ^ Ind^i^)- Ind(j(5n {Ф ^ c}) ~ m + 1.

Proof It is enough to apply the above proposition with D = 0 (i.e. regular index), C^B,<zBшdF-={ф^c}.