KO-rings and J-groups of lens spaces

Abstract

In this thesis, we make the explicit computation of the real A'-theory of lens spaces and making use of these results and Adams conjecture, we describe their .7-groups in terms of generators and relations. These computations give nice by-products on some geometrical problems related to lens spaces. We show that J-groups of lens spaces approximate localized J-groups of complex projective spaces. We also make connections of the J-cornputations with the classical cross-section problem and the .James numbers conjecture. Many difficult geometric problems remain open. The results are related to some arithmetic on representations of cyclic groups o\er fields and the Atiyah-Segal isomormhisrn. Eventually, we are interested in representations over rings, in connection with Algebraic K-theory. This turns out to lie a very non-trivial arithmetic problem related to number theory.