Ações de Zr2 fixando RPj U CPk

Abstract

The classification up to equivariant cobordism of smooth involutios (M, T) having fixed set F is a classical problem in cobordism theory. This classification has been studied for several cases of F, of which we highlight the following: For F = RPj, the j-dimensional real projective space, the classification was established by P. E. Conner, E. E. Floyd and R. E. Stong in [6] and [26]. In [24], D. C. Royster studied this problem with F = RPj U RPk, for naturals numbers j and k, except when j and k are both even and greater than zero. R. Oliveira, P. L. Q. Pergher and A. Ramos established the classification for F = RPj U RPk where j = 2 and k is even in [17]. The general case where j and k are both even and greater than zero is still open. For F = CPj and F = HPj, where CPj and HPj are the corresponding complex and quaternionic projective spaces, the classification was established by P. L. Q. Pergher and A. Ramos in [21]. They also established the classification for F = CPj U CPk and F = HPj U HPk, except when j and k are both even and greater than zero, but they resolved this problem for the particular case j = 2* and k even. As in the real case, also for complex and quaternionic projective spaces, the general case where j and k are both even and greater than zero is still open. In this work we deal with the classification, up to equivariant cobordism, of the pairs (M, T) for which the fixed point set is F = RPj U CPk, including the “hard”case where j and k are both even and greater than zero. We also extend the classification for Z^-actions in the case that both dimensions are even.