Involuções fixando RP(6)URP(2n) e variedades compatíveis com o ponto com respeito à involuções

Abstract

In this work, we have two objectives: the first lives in the context of the classification, up to equivariant cobordism, of the pairs (M, T) , where M is a closed and smooth manifold and T is a smooth involution defined in M , with a prefixed fixed point set F . This is a well-established problem in the literature, and for reasons that will be explained in the introduction of this work, an important case is when F is an union of real projective spaces. Concerning this case, for F = R P(n) , such a classification was determined by P. E. Conner and E. E. Floyd, for n odd, and by R. E. Stong for n even. D. C. Royster determined such a classification when F is the disjoint union of two real projective spaces, F = R P(m)UR P(n) , but he left open the cases where m and n are even and greater than zero. P. L. Q. Pergher and A. Ramos worked on such open cases, solving the particular cases in which m is a power of 2 and n> 0 is any even natural number. Thus, taking into account the works of Royster, P. Pergher and A. Ramos, the first open case was F = R P(6) UR P(2n) . In our work, we obtain the classification for this open case; furthermore, we extend it to pairs (M, \Phi) , where \Phi is an smooth action of the group (Z_2)^k in M , where (Z_2)^k is understood here as the group generated by k commuting involutions T_1 , T_2 ,...., T_k defined in M .

Our second objective is to deal with a definition, created by us, related to a certain property associated with a closed, connected and smooth manifold. Let F be such a manifold. We say that F satisfies the property CP ( compatible with the point) if there exists a closed and smooth manifold M and a smooth involution T such that the fixed point set of T is F U{point } . The inspiration for this definition was the fact that, Conner and Floyd proved that, among the spheres S ^ n , the only ones that satisfy such property were S ^ 1 , S ^ 2 , S ^ 4 and S ^ 8 , and later, P. Pergher determined all products of spheres that satisfy this property. Firstly, we determine some simple results of validity and non-validity of CP , among which we highlight the following intriguing result: every manifold of dimension 1 , 2 , 4 or 8 satisfies CP . However, the most intricate part of our work was some results of non validity of the property CP for Dold manifolds P (m, n) .