  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.
