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These are notes, by Z. Fiedorowicz, from lectures given by J. Frank Adams atthe University of Chicago in spring of 1973. They give an elegant axiomaticpresentation of localization and completion in algebraic topology. Theconstruction of localization and completion functors with respect to anarbitrary generalized homology theory is derived from the axioms by using theBrown representability theorem. These notes were never formally published, dueto an apparent flaw in the proof. The relevant representable functors could notbe shown to be set-valued, as opposed to class-valued. Subsequent work by A. K.Bousfield established the existence of these functors, using more technicalsimplicial methods. These functors are now an essential tool in homotopytheory. The notes also contain an addendum devoted to establishing that acertain element in the gamma family of the stable homotopy groups of spheres isnonzero, using Brown-Peterson homology. At that time this was a matter ofcontroversy, as S. Oka and H. Toda claimed to have proved the contrary result.  Besides being of historical interest, these notes give a very readableintroduction to localization and completion, with minimal prerequisites. Abrief epilogue by Z. Fiedorowicz fills the gap in the proof and sketches somefollow up history. A foreword and a few additional editorial notes have alsobeen added.

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