These are the terse notes for a graduate seminar which I conducted at Harvard during the Fall of 1963. By and large my audience was acquainted with the standard material in bundle theory and algebraic topology and I therefore set out directly to develop the theory of characteristic classes in both the standard cohomology theory and K-theory. Since 1963 great strides have been made in the study of K(X), notably by Adams in a series of papers in Topology. Several more modern accounts of the subject are available. In particular the notes of Atiyah, IINotes on K-theoryll not only start more elementarily, but also carry the reader further in many respects. On the other hand, those notes deal only with K-theory and not with the characteristic vii 406 viii classes in the standard cohomology. The main novelty of these lectures is really the systematic use of induced representation theory and the resulting formulae for the KO-theory of sphere bundles. Also my point of view toward the J -invariant, e(E) is slightly different from that of Adams. I frankly like my groups H\ Z+; KO(X)) and there is some indication that the recent work of Sullivan will bring them into their own.

This volume features the collected papers of Raoul Bott. It contains his work on mathematics related to physics from the 1980s.