Université de Paris, France
The periodicity of Clifford algebras noticed by Atiyah, Bott and Shapiro led to considerable developments in K-theory during the 60's. We reconsider this thema by giving an explicit formula for the equivariant K-theory of real projective bundles. This formula implies the classical computation by J.-F. Adams of the K-theory of real projective spaces, Bott's exact sequence relating real and complex K-theory, etc., together with an unexpected relation between various versions (due to Atiyah) of equivariant K-theory . This last result was used recently in a new computation of the Witt group of real curves by Schlichting, Weibel and the author.