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 also 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 also show that we can lift K-theory with Reality to a Z/2-spectrum KR, and prove 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.