Teorida de G-índice e grau de aplicações G-equivariantes

Abstract

Before the appearance of the paper “An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theoremsof Fadell and Husseini [20], had been considered numerical indices of G -spaces, when G =Z2 and when G is a finite group. However, such numerical indices are obviously insufficient in the case of groups more complexes, for example, G =S1. In this context Fadell and Husseini, introduced the called valued-ideal cohomological index: to every paracompact G -space X they associated an ideal Ind (X ,K) of the cohomology ring Ȟ ∗(BG ;K), where the Cech cohomology Ȟ ∗ is considered with coefficients in a fieldK and BG is the classifying space of the group G . Moreover, they associated to this ideal the numerical valued cohomological index, that is, the dimension ofK-vector space obtained by the quotient between the ring Ȟ ∗(BG ;K) and the ideal Ind (X ,K). The main objective of this work is to present a detailed study of this index and use such index on the study of results on degree of equivariant maps proved by Hara in his paper “The degree of equivariant maps[24]. keywords: G -spaces, G -equivariant maps, classifying spaces, G -index, Cech cohomology.