Involuções fixando somas conexas de espaços projetivos e melhorias para o Five Halves Theorem quando Fix(T ) = F n ∪ F j

Abstract

Let M m be a closed and smooth m-dimensional manifold, and T : M m → M m a smooth involution, that is, a period 2 diffeomorphism defined on M m . It is well known the fact that the fixed point set of T , F = {x ∈ M m |T (x) = x}, is a finite and disjoint union of closed smooth submanifolds, whose dimensions can vary from 0 to m. We write F = ∪ni=0 F i , n ≤ m, where F i denotes the disjoint union of the i-dimensional components of F. The famous Five Halves Theorem of J. Boardman assures that, if the pair (M m , T ) does not bound equivariantly, then we have m ≤ 52 n, and with this level of generality, this bound is best possible. This result motivated P. Pergher to introduce, in the literature, the following type of question: is it possible to improve the Boardman’s bound by imposing the omission of some dimensions of F? In this work, our first objective is obtaining a result of this type, specifically the case where F has the form F n ∪ F j , 0 ≤ j < n, and with F n ∪ F j not being a boundary. There are several results of this nature in the literature, as will be detailed in the Introduction in historical and chronological terms. The second goal of this work lives in the context of classifying, up to equivariant cobordism, involutions (M, T ) whose fixed point set is a pre-selected manifold (or a disjoint union of manifolds) F. This line of problems is well-established in the literature, see in the Introduction references with several correlated results. Specifically, in this work we will address the case in which F is a connected sum of two projectives spaces real, complex and quaternionic, F = Kd P(n)#Kd P(n), with n odd, where d = 1, 2 and respectively symbolize the real, complex and quaternionic cases. Again, the relationship of this case with existing cases in the literature will be described in the Introduction.