The Higher K-Theory of Real Curves

Abstract

Abstract. If X is a smooth curve defined over the real numbers R, we show that Kn(X) is the sum of a divisible group and a finite elementary abelian 2-group when n ≥ 2. We determine the torsion subgroup of Kn(X), which is a finite sum of copies of Q/Z and Z/2, only depending on the topological invariants of X(R) and X(C), and show that (for n ≥ 2) these torsion subgroups are periodic of order 8. Let X be a real curve, i.e., a smooth curve X which is defined over R but not over C. Classical calculations dating to 1883 ([Whd]; see §1) clearly show the relation between the topological invariants of X(R) and the Picard group (and hence K0) of X. In this paper we shall show that there is a strong connection between the higher algebraic K-theory of X and the topological invariants of the underlying space X(R) of its real points, equipped with the Euclidean topology. The link between these will be forged from recent developments in motivic cohomology. When X is projective, we have the following satisfactory answer. Main Theorem 0.1. Let X be an irreducible, smooth projective real curve such that X(R) has ν> 0 components. Then the groups Kn(X) are the direct sum of a divisible group and an elementary abelian 2-group for n ̸ = 0. Moreover, the torsion subgroups Kn(X)tors are periodic of period 8. In fact, if X has genus g then for n ≥ 0: Kn(X)tors ∼ =