THE METHOD OF INFINITE DESCENT IN STABLE HOMOTOPY THEORY I January 29, 2001

This paper is the first in a series aimed at clarifying and extending of parts of the last chapter of [Rav86], in which we described a method for computing the AdamsNovikov 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 know 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 [Rav86, 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. The method referred to in the title involves the connective p-local ring spectra T (m) of [Rav86, §6.5], which satisfy