K-theory with Reality

Abstract

K-theory is a classical algebraic invariant for compact Hausdorff spaces constructed out of complex or real vector bundles. In 1966, M. Atiyah came up with a slightly different version for spaces with a group action of Z/2, called K-theory with Reality, that allows to recover both real and complex K-theory. We discuss the highlights of this theory and see that it forms a RO(Z/2)-graded cohomology theory. Moreover, we present a detailed exposition of stable homotopy theory and equivariant stable homotopy theory, introducing spectra and their equivariant analogues G-spectra from a categorical perspective, where G is a finite group. We show that we can lift K-theory with Reality to a Z/2-spectrum KR. The main goal of this thesis is to show that the homotopy fixed points of KR is isomorphic, in the stable homotopy category, to the spectrum KO of real K-theory, avoiding the computation of Dugger's homotopy fixed points spectral sequence.