On the k-invariants of iterated loop spaces

SynopsisThe purpose of this paper is to give universal bounds for the order of the Postnikov k-invariants of infinite loop spaces. This is done by giving universal bounds for the order of the k-invariants of m-connected r-fold loop spaces in dimensions ≦ r + 2m. An application of the result provides information on the Hurewicz homomorphism between the algebraic K-theory of aring and the homology of its general linear group.</jats:p

Universal bounds for the exponent of stable homotopy groups

AbstractLet X be an m-connected CW-complex and n an integer satisfying 2⩽n⩽2m. We prove that if the nth integral homology group of X is of finite exponent, then the nth homotopy group of X has also finite exponent, and we give a universal bound for this exponent. This provides for instance universal bounds for the exponent of the stable homotopy groups of Moore spaces

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