Duas involuções comutantes fixando certas variedades de Dold e certas uniões de espaços projetivos relativos a anéis diferentes

Abstract

Let (M; _) be a smooth action of Z2 2 on a closed, smooth and m-dimensional manifold, with fixed point set F = m[ j=0 Fj , where Fj means the union of the components of F of dimension j. Given such a F, we can ask for the actions that have F as fixed point set, and in the positive case we have the question of the cobordism classification of such actions. In this work we obtain such classifications when F is a Dold manifold of the form P(1; 2n + 1), and when F is the disjoint union of two projectives spaces of the form KdP(m) [ KeP(2n + 1), with d; e 2 f1; 2; 4g and d < e. In particular, this last case is concerning questions leaved open in [2]. The crucial point concerning such classifications was an improvement refering to the Pergher s result ([22], Theorem 1), which established conditions on a collection of three vector bundles in such a way it can be realized as the fixed data of some Z2 2-action. Specifically, we become such conditions more efficient in computational terms, by removing one of the conditions, specifically the more complicated. This possibilited to detect certain exotic actions (see the definition 1.9.15) in the mentioned classifications.