Annals of Mathematics Studies Number 113 ALGEBRAIC TOPOLOGY AND ALGEBRAIC K-THEORY EDITED BY WILLIAM BROWDER Proceedings of a conference October 24-28, 1983, at Princeton University, dedicated to John C. Moore on his 60th birthday PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1987 Copyright © 1987 by Princeton University Press ALL RIGHTS RESERVED The Annals of Mathematics Studies are edited by William Browder, Robert P. Langlands, John Milnor, and Elias M. Stein Corresponding editors: Stefan Hildebrandt, H. Blaine Lawson, Louis Nirenberg, and David Vogan Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding materials are chosen for strength and durability. Pa­ perbacks, while satisfactory for personal collections, are not usually suitable for library rebinding ISBN 0-691-08415-7 (cloth) ISBN 0-691-08426-2 (paper) Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey ☆ Library of Congress Cataloging in Publication data will be found on the last printed page of this book DEDICATION TO JOHN MOORE John Moore, in his mathematical career and his more than 30 years at Princeton, has made a lasting impact on the subject of topology, both in his research and in his influence on the many graduate students who worked with him. This conference and this volume are dedicated to him in recognition and warm appreciation by his ex-students, by other coworkers in topology, and by his many friends, numbered in both groups. Students of John Moore Richard Swan 1957 William Browder 1958 James Stasheff 1961 Allan Clark 1961 Stephen Weingram 1962 Paul Baum 1962 Wu-yi Hsiang 1964 J. Peter May 1964 William Singer 1967 Harsh V. Pittie 1970 Fred William Roush 1972 Harold Hastings 1972 Joseph Neisendorfer 1972 Brian Smith 1972 Philip Trauber 1973 James Lin 1974 Haynes Miller 1975 Leo Chouinard 1975 Robert Thomason 1977 Paul Selick 1977 John Long 1979 v LIST OF SPEAKERS AT THE CONFERENCE J. F. Adams D. Anick E. H. Brown G. Carlsson F. Cohen R. Cohen W. Dwyer E. Friedlander S. Halperin D. Kan J. Lin M. Mahowald H. Miller J. C. Moore J. Neisendorfer F. P. Peterson D. Ravene1 P. Selick J.-P. Serre W. Singer C. Soule J. Stasheff R. Swan R. Thomason F. Waldhaussen A. Zabrodsky CONTENTS Preface ix I EXPONENTS IN HOMOTOPY THEORY 3 by F. R. Cohen, J. C. Moore, and J. A. Neisendorfer II THE EXPONENT OF A MOORE SPACE 35 by Joseph A. Neisendorfer III THE SPACE OF MAPS OF MOORE SPACES INTO SPHERES 72 by H. E. A. Campbell, F. R. Cohen, F. P. Peterson, and P. S. Selick IV THE ADAMS SPECTRAL SEQUENCE OF Q2S3 AND BROWN- GITLER SPECTRA 101 by Edgar H. Brown and Ralph L. Cohen V HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES 126 by Donald M. Davis and Mark Mahowald VI MAPPING TELESCOPES AND K^-LOCALIZATION 152 by Donald M. Davis, Mark Mahowald and Haynes Miller VII THE GEOMETRIC REALIZATION OF THE CHROMATIC RESOLUTION 168 by Douglas C. Ravenel VIII EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS 180 by W. G. Dwyer and D. M. Kan IX THE ROLE OF THE STEENROD ALGEBRA IN THE MOD 2 COHOMOLOGY OF A FINITE H-SPACE 206 by James P. Lin X MAPS BETWEEN CLASSIFYING SPACES 228 by A. Zabrodsky XI GENERIC ALGEBRAS AND CW COMPLEXES 247 by David J. Anick vii viii CONTENTS XII DEFORMATION THEORY AND THE LITTLE CONSTRUCTIONS OF CARTAN AND MOORE 322 by James Stasheff XIII FREE (Z2)3 - ACTIONS ON FINITE COMPLEXES 332 by Gunnar CarIsson XIV EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA 345 by J. P. May XV A DECOMPOSITION OF THE SPACE OF GENERALIZED MORSE FUNCTIONS 365 by Ralph L. Cohen XVI ALGEBRAIC K-THEORY OF SPACES, CONCORDANCE AND STABLE HOMOTOPY THEORY 392 by Friedhelm Waldhausen XVII THE MAP BSG -> A(*) -» QS° 418 by Marcel Bokstedt and Friedhelm Waldhausen XVIII VECTOR BUNDLES, PROJECTIVE MODULES AND THE K-THEORY OF SPHERES 432 by Richard G. Swan XVIX LIMITS OF INFINITESMAL GROUP COHOMOLOGY 523 by Eric M. Friedlander and Brian J. Parshall XX ALGEBRAIC K-THEORY OF GROUP SCHEME ACTIONS 539 by R. W. Thomason PREFACE From October 24 to October 28, 1983, a conference entitled 'Algebraic Topology and Algebraic K-theory’ was held at Princeton University, and was dedicated to John Moore on his 60th birthday. It was planned by an organizing committee consisting of William Browder, chairman, Franklin P. Peterson, James Stasheff and Richard Swan. It was the intention of the organizers to organize a conference that emphasized homotopy theory, algebraic topology more generally and some more topologically related aspects of algebraic K-theory, and to create an interesting conference with a relatively narrow focus, without trying to cover all aspects of topology, or even of algebraic topology. Neither was it the intention of the committee to try to reflect explicitly the interests of Moore, or to emphasize explicitly his role and that of his students in so many of the modern developments. In the event, these things were, in fact, well displayed during the conference. The conference was part of a special year in Algebraic Topology and K-theory at Princeton University during the 1983-84 academic year, which was supported by the National Science Foundation. Special thanks are due to Nancy Schlesinger who handled most of the day to day organization of the conference, as well as to a large number of Princeton graduate students who pitched in to help with refreshments, and other aspects of hospitality for the visitors. W. Browder ix