Estimativas ótimas para certos teoremas generalizados de Borsuk-Ulam e Ljusternik-Schnirelmann

Abstract

The classic Theorems of Borsuk-Ulam and Ljusternik-Schnirelmann have many generalizations, among which we point out that given by C. Schupp [12] and H. Steinlein [14]. Schupp generalizes the Borsuk-Ulam Theorem by replacing the Z2-free action on the n-sphere by a Zp-free action, where p is any prime number. In the generalization of the Ljusternik-Schnirelmann Theorem maden by Steinlein, the n-sphere is replaced by a normal space M on which Zp acts freely. We explore in this dissertation the subsequent results of Steinlein [15] in which is proved that the estimates of the Schupp s Theorem are the best possible and the estimates for the Steinlein s Theorem can be improved in certain cases, furthermore a sort of converse of the Steinlein Theorem is valid. The concept of genus of a Zp-space is fundamental for these theorems and the genus of the n-sphere is n + 1 independently of the prime number and the Zp-free action on Sn. We realize that the method employed in the proof on this result can be used to estimate an upper bound for the genus of a topological n-manifold that admits a Zp-free action.