k-invariants for K-theory of curves over global fields

Dominique Arlettaz, Grzegorz Banaszak
2009 Journal of K-Theory  
Introduction. Let X be a connected simple CW-complex and let α n : X → X[n] denote the n-th Postnikov section of X for any positive integer n : X[n] is the CW-complex obtained from X by killing the homotopy groups of X in dimensions > n, more precisely by adjoining cells of dimensions ≥ n + 2 such that π j (X[n]) = 0 for j > n and (α n ) * : π j (X) → π j (X[n]) is an isomorphism for j ≤ n. The Postnikov k-invariants of X are cohomology classes k n+1 (X) ∈ H n+1 (X[n − 1]; π n (X)), for n ≥ 2,
more » ... n (X)), for n ≥ 2, which provide the necessary information for the reconstruction of X, up to a weak homotopy equivalence, from its homotopy groups (see for instance [W], Section IX.2 for a definition). In this paper we extend our results of [AB] concerning k-invariants to the case of algebraic curves over global fields. In section 2 we state general results about kinvariants and we discuss the cases when they are of finite order. Then in section 3 we consider the case of k-invariants of Quillen andÉtale K-theory spaces and we explain why they are of finite order. In section 4 we prove our main result:
doi:10.1017/is008012021jkt071 fatcat:4odfbuoqjncnbnl54vulrsz4pa