The Method of Infinite Descent in Stable

This paper is a continuation of [Rav02] and the second in a series of papers intended to clarify and expand results of the last chapter of [Rav04], in which we described a method for computing the Adams-Novikov E2-term and used it to determine the stable homotopy groups of spheres through dimension 108 for p = 3 and 999 for p = 5. The latter computation was a substantial improvement over prior knowledge, and neither has been improved upon since. It is generally agreed among homotopy theorists that it is not worthwhile to try to improve our knowledge of stable homotopy groups by a few stems, but that the prospect of increasing the known range by a factor of p would be worth pursuing. This possibility may be within reach now, due to a better understanding of the methods of [Rav04] Chapter 7 and improved computer technology. This paper should be regarded as laying the foundation for a program to compute π∗(S) through roughly dimension p|v2|, i.e., 432 for p = 3 and 6,000 for p = 5. We will refer to results from [Rav02] freely as if they were in the first four sections of this paper, which begins with Section 5.