Algebraic and Real K-theory of Real Varieties, by Max Karoubi and Charles Weibel

Let V be a smooth variety defined over the real numbers. Every algebraic vector bundle on V induces a complex vector bundle on the underlying topological space V(C), and the involution coming from complex conjugation makes it a Real vector bundle in the sense of Atiyah. This association leads to a natural map from the algebraic K-theory of V to Atiyah's ``Real K-theory'' of V(C).

Passing to finite coefficients Z/m, we show that the maps from K_n(V;Z/m) to KR^-n(V(C);Z/m) are isomorphisms when n is at least the dimension of V, at least when m is a power of two. Our key descent result is a comparison of the K-theory space of V with the homotopy fixed points (for complex conjugation) of the K-theory space of the complex variety V_C.

When V is the affine variety of the d-sphere, it turns out that KR^*(V(C))=KO^*(S^d). In this case we show that for all nonnegative n we have

K_n(V;Z/m) = KO^-n(S^d;Z/m).

This paper has appeared in Topology 42 (2003), 715-742.

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