The homotopy groups of the algebraic K -theory of the sphere spectrum

We calculate pi K-*(S) circle times Z[1/2], the homotopy groups of K(S) away from 2, in terms of the homotopy groups of K(Z), the homotopy groups of CP-1 infinity and the homotopy groups of S. This builds on work of Waldhausen, who computed the rational homotopy groups (building on work of Quillen and Borel) and Rognes, who calculated the groups at odd regular primes in terms of the homotopy groups of CP-1 infinity and the homotopy groups of S.