Localisation and Completion with an addendum on the use of Brown-Peterson homology in stable homotopy

TL;DR

This paper provides an axiomatic approach to localization and completion in algebraic topology, discusses the use of Brown-Peterson homology to establish nontrivial elements in stable homotopy groups, and offers historical insights into these topics.

Contribution

It presents an elegant axiomatic framework for localization and completion functors in algebraic topology, and includes an addendum on Brown-Peterson homology's role in stable homotopy groups.

Findings

Established that a certain element in the gamma family is nonzero using Brown-Peterson homology.

Provided an accessible introduction to localization and completion with minimal prerequisites.

Included historical context and a correction to earlier proof gaps.

Abstract

These are notes, by Z. Fiedorowicz, from lectures given by J. Frank Adams at the University of Chicago in spring of 1973. They give an elegant axiomatic presentation of localization and completion in algebraic topology. The construction of localization and completion functors with respect to an arbitrary generalized homology theory is derived from the axioms by using the Brown representability theorem. These notes were never formally published, due to an apparent flaw in the proof. The relevant representable functors could not be shown to be set-valued, as opposed to class-valued. Subsequent work by A. K. Bousfield established the existence of these functors, using more technical simplicial methods. These functors are now an essential tool in homotopy theory. The notes also contain an addendum devoted to establishing that a certain element in the gamma family of the stable homotopy groups of spheres is nonzero, using Brown-Peterson homology. At that time this was a matter of controversy, as S. Oka and H. Toda claimed to have proved the contrary result. Besides being of historical interest, these notes give a very readable introduction to localization and completion, with minimal prerequisites. A brief epilogue by Z. Fiedorowicz fills the gap in the proof and sketches some follow up history. A foreword and a few additional editorial notes have also been added.