Adams Memorial Symposium on Algebraic Topology: Volume 1 by Nigel Ray, Grant Walker

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 used to be devoted to his reminiscence, and almost all the world's best specialists took half. This two-volume paintings constitutes the court cases of the symposium. The articles contained right here diversity from overviews to experiences of labor nonetheless in growth, in addition to a survey and whole bibliography of Adams' personal paintings. those complaints shape an incredible compendium of present study in algebraic topology, and one who demonstrates the intensity of Adams' many contributions to the topic. the following within the first quantity the subject is principally volatile homotopy concept, homological and express algebra. the second one quantity is orientated towards solid homotopy concept, the Steenrod algebra and the Adams spectral series.

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Adams left open the question of realizability of the last two, but that was settled in the negative in John Hubbuck's thesis. 3. Finite H-spaces and algebras over the Steenrod algebra (1978, with Clarence Wilkerson) [63] Explicit necessary and sufficient algebraic conditions are given on an unstable A-algebra P which ensure that P - H*(BT)w for a torus T and finite group W. These conditions hold if P is a polynomial algebra on generators of even degree prime to p, and W is then a p-adic generalized reflection group; moreover, there is a space X such that P - H*(X).

F. Finite H-spaces and compact Lie groups 1. The sphere considered as an H-space mod p (1960) [16] The very first appearance of the localization of a space appeared here, presaging the study of finite H-spaces via localization. Adams gave the slogan "an odd sphere is an H-space mod p" as early as 1956. 2. H-spaces with few cells (1960) [21] It is shown that if H*(G; Z2) = E(xq, x,yy), q < n, then q and n are both one of 1, 3, 7, or else (q, n) = (1,2), (3, 5), (7,11), or (7,15). Adams left open the question of realizability of the last two, but that was settled in the negative in John Hubbuck's thesis.

It contains capsule introductions to a variety of topics, none of which have yet made it to any text book. The exposition is delightful, despite the elephantine jokes. "I am very grateful to J. P. May, B. J. Sanderson and S. B. Priddy for reading the first draft of this book . . ; I have benefited greatly from their comments. " It was deliberately thin on details, except for its discussion of the transfer and its quite full treatment of the Madsen, Snaith, Tornehave theorem that passage to maps of zeroth spaces gives an injection [bso, bso] -> [BSO, BSO].

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