By Nigel Ray, Grant Walker
J. Frank Adams had a profound impression on algebraic topology, and his works proceed to form its improvement. The overseas Symposium on Algebraic Topology held in Manchester in the course of July 1990 was once devoted to his reminiscence, and nearly the entire world's best specialists took half. This two-volume paintings constitutes the court cases of the symposium. The articles contained the following diversity from overviews to stories of labor nonetheless in growth, in addition to a survey and whole bibliography of Adams' personal paintings. those court cases shape a tremendous compendium of present study in algebraic topology, and one who demonstrates the intensity of Adams' many contributions to the topic. right here within the first quantity the topic is principally volatile homotopy idea, homological and specific algebra. the second one quantity is orientated towards solid homotopy conception, the Steenrod algebra and the Adams spectral series.
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Additional info for Adams Memorial Symposium on Algebraic Topology: Volume 1
Brown's representability theorem (1970)  A group-valued contravariant functor on the homotopy category of finite CW-complexes is shown to be representable if it satisfies the wedge and Mayer-Vietoris axioms. Brown showed this for functors taking countable sets as values; thus the result eliminates the annoying countability assumption at the price of insisting on group values. A consequence is that any homology theory which satisfies the direct limit axiom is representable. 4. Idempotent functors in homotopy theory (1973)  The notion of an idempotent functor is advertised in connection with localization and completion theory.
May) A generalisation of the Atiyah-Segal completion theorem, Topology 27 (1988), 1-6. Bibliography of J. F. Adams 27 80. (with N. J. Kuhn) Atomic spaces and spectra, Proc. Edinburgh Math. Soc. (2) 32 (1989), 473-481. 81. (with Z. Wojtkowiak) Maps between p-completed classifying spaces, Proc. Roy. Soc. Edinburgh 112A (1989), 231-235. 82. Talk on Toda's work, in Homotopy theory and related topics, Lecture Notes in Math. 1418, Springer 1990, 7-14. 1990. 83. Two theorems of J. Lannes, Collected Works, Cambridge Univ.
For any r, the real e invariant detects Im J as a direct summand, the case r - 7 mod 8 being incomplete in the paper since the full Adams conjecture was not yet available. Various composition products and Toda brackets are detected by means of Yoneda and Massey products in the target groups. This seems a little like a magical boot strap operation since the Horn and Ext calculations involved are fairly elementary. The conception is a marvelous example of algebraic modeling of topological phenomena.