On the algebraic K-theory of Z

AbstractLet GL(Z) (respectively SL(Z)) be the infinite general (respectively special) linear group and St(Z) the infinite Steinberg group of Z. This paper studies the relationships between KiZ ≔ μiBGL(Z)+, Hi(SL(Z); Z) and Hi(St(Z); Z) for i = 4 and 5 (they are well understood for i ≤ 3). The main results describe the Hurewicz homomorphism KiZ → Hi(St(Z); Z): it is an isomorphism if i = 4 and its cokernel is cyclic of order 2 if i = 5 (more precisely, the induced homomorphism K5Z/torsion → H5(St(Z); Z)/torsion is multiplication by 2). The relations between the integral homology of St(Z) and that of SL(Z) in dimensions 4 and 5 are also explained