Determining the algebraic K-theory of rings of integers in number fields has been the goal of much research. In [10] D. Quillen showed that the Hurewicz map h : Q0(S ) → BGL(Z) (see 1.1 for the notation) induces an interesting map on homotopy groups from the stable homotopy groups of spheres to the algebraic K-theory of the ring Z of rational integers. Quillen observed that if ` is an odd prime...

As is well known, the image of J-homomorphism in the stable homotopy groups of spheres is described in terms of the first line of Adams-Novikov E2-term. In this paper we consider an algebraic analogue of the image J using the spectrum T (m)(j) defined by Ravenel and determine the Adams-Novikov first line for small values of j.

We define fiber bundles and discuss the long exact sequence of homotopy groups of a fiber bundle, and we give the Hopf bundles as examples. We also prove the Freudenthal suspension theorem for spheres. All results are applied immediately to homotopy groups of spheres.

We prove that for many degrees in a stable range the homotopy groups of the moduli space of metrics of positive scalar curvature on S and on other manifolds are non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov-Lawson to an exotic smooth families of spheres due to Hatcher.

Kervaire and Milnor's germinal paper [15], in which they used the newly-discovered techniques of surgery to begin the classification of smooth closed manifolds homotopy equivalent to a sphere (homotopy-spheres), was intended to be the first of two papers in which this classification would be essentially completed (in dimensions > 5). Unfortunately , the second part never appeared. As a result, ...