An Atiyah–Hirzebruch Spectral Sequence for K R-Theory

Abstract

In recent years much attention has been given to a certain spectral sequence relating motivic cohomology to algebraic K-theory [Be, BL, FS, V3]. This spectral sequence takes on the form H(X,Z(− q 2 )) ⇒ K(X), where the H(X ;Z(t)) are the bi-graded motivic cohomology groups, and K(X) denotes the algebraic K-theory of X . It is useful in our context to use topologists’ notation and write K(X) for what K-theorists call K−n(X). The above spectral sequence is the analog of the classical Atiyah-Hirzebruch spectral sequence relating ordinary singular cohomology to complex K-theory, in a way that is explained further below. It is well known that there are close similarities between motivic homotopy theory and the equivariant homotopy theory of Z/2-spaces (cf. [HK1, HK2], for example). In fact there is even a forgetful map of the form (motivic homotopy theory over R) → (Z/2-equivariant homotopy theory),