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Contemporary Mathematics 239 CONTEMPORARY MATHEMATICS 239 Homotopy Invariant Algebraic Structures A Conference in Honor of J. Michael Boardman AMS Special Session on Homotopy Theory January 7-10, 1998 Baltimore, MD Jean-Pierre Meyer Jack Morava W. Stephen Wilson Editors http://dx.doi.org/10.1090/conm/239 Selected Titles in This Series 239 Jean-Pierre Meyer, Jack Morava, and W. Stephen Wilson, Editors, Homotopy invariant algebraic structures, 1999 238 Gui-Qiang Chen and Emmanuele DiBenedetto, Editors, Nonlinear partial differential equations, 1999 237 Thomas Branson, Editor, Spectral problems in geometry and arithmetic, 1999 236 Bruce C. Berndt and Fritz Gesztesy, Editors, Continued fractions: From analytic number theory to constructive approximation, 1999 235 Walter A. Carnielli and Itala M. L. D'Ottaviano, Editors, Advances in contemporary logic and computer science, 1999 234 Theodore P. Hill and Christian Houdre, Editors, Advances in stochastic inequalities, 1999 233 Hanna Nencka, Editor, Low dimensional topology, 1999 232 Krzysztof Jarosz, Editor, Function spaces, 1999 231 Michael Farber, Wolfgang Liick, and Shmuel Weinberger, Editors, Tel Aviv topology conference: Rothenberg Festschrift, 1999 230 Ezra Getzler and Mikhail Kapranov, Editors, Higher category theory, 1998 229 Edward L. Green and Birge Huisgen-Zimmermann, Editors, Trends in the representation theory of finite dimensional algebras, 1998 228 Liming Ge, Huaxin Lin, Zhong-Jin Ruan, Dianzhou Zhang, and Shuang Zhang, Editors, Operator algebras and operator theory, 1999 227 John McCleary, Editor, Higher homotopy structures in topology and mathematical physics, 1999 226 Luis A. Caffarelli and Mario Milman, Editors, Monge Ampere equation: Applications to geometry and optimization, 1999 225 Ronald C. Mullin and Gary L. Mullen, Editors, Finite fields: Theory, applications, and algorithms, 1999 224 Sang Geun Hahn, Hyo Chul Myung, and Efim Zelmanov, Editors, Recent progress in algebra, 1999 223 Bernard Chazelle, Jacob E. Goodman, and Richard Pollack, Editors, Advances in discrete and computational geometry, 1999 222 Kang-Tae Kim and Steven G. Krantz, Editors, Complex geometric analysis in Pohang, 1999 221 J. Robert Dorroh, Gisele Ruiz Goldstein, Jerome A. Goldstein, and Michael Mudi Tom, Editors, Applied analysis, 1999 220 Mark Mahowald and Stewart Priddy, Editors, Homotopy theory via algebraic geometry and group representations, 1998 219 Marc Henneaux, Joseph Krasil'shchik, and Alexandre Vinogradov, Editors, Secondary calculus and cohomological physics, 1998 218 Jan Mandel, Charbel Farhat, and Xiao-Chuan Cai, Editors, Domain decomposition methods 10, 1998 217 Eric Carlen, Evans M. Harrell, and Michael Loss, Editors, Advances in differential equations and mathematical physics, 1998 216 Akram Aldroubi and EnBing Lin, Editors, Wavelets, multiwavelets, and their applications, 1998 215 M. G. Nerurkar, D. P. Dokken, and D. B. Ellis, Editors, Topological dynamics and applications, 1998 214 Lewis A. Coburn and Marc A. Rieffel, Editors, Perspectives on quantization, 1998 213 Farhad Jafari, Barbara D. MacCluer, Carl C. Cowen, and A. Duane Porter, Editors, Studies on composition operators, 1998 (Continued in the back of this publication) Homotopy Invariant Algebraic Structures J. Michael Boardman CoNTEMPORARY MATHEMATICS 239 Homotopy Invariant Algebraic Structures A Conference in Honor of J. Michael Boardman AMS Special Session on Homotopy Theory January 7-10, 1998 Baltimore, MD Jean-Pierre Meyer Jack Morava W. Stephen Wilson Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor Andreas Blass Andy R. Magid Michael Vogelius The Proceedings of a series of Special Sessions on Homotopy Theory, held at the Annual Meeting of the American Mathematical Society in Baltimore, MD, January 7-10, 1998. Letter from Rene Thorn toW. Stephen Wilson used with permission. 1991 Mathematics Subject Classification. Primary 55-03, 55Nxx, 55Pxx, 55Qxx, 55Txx. Library of Congress Cataloging-in-Publication Data AMS Special Session on Homotopy Theory (1998: Baltimore, Md.) Homotopy invariant algebraic structures : a conference in honor of J. Michael Boardman : AMS Special Session on Homotopy Theory, January 1998, Baltimore, MD / Jean-Pierre Meyer, Jack Morava, W. Stephen Wilson, editors. p. em. -(Contemporary mathematics, ISSN 0271-4132; 239) Includes bibliographical references. ISBN 0-8218-1057-X (alk. paper) 1. Homotopy theory-Congresses. I. Boardman, J. M. (John Michael). II. Meyer, Jean- Pierre, 1929- . III. Morava, Jack, 1944- . IV. Wilson, W. Stephen, 1946- V. Title. VI. Series: Contemporary mathematics (American Mathematical Society) ; v. 239. QA612.7.A57 1998 514'.24-dc21 99-34575 CIP Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionlllams. org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 1999 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http: I /www. ams. org/ 10987654321 04 03 02 01 00 99 Contents Preface IX Publications of J. Michael Boardman xi Some History Letter from R. Thorn 3 Higher homotopies, Pacts, and the bar construction S. MAC LANE 5 The hare and the tortoise J.P. MAY 9 Cobordism of involutions revisited, revisited J. MORAVA 15 Grafting Boardman's cherry trees to quantum field theory J. STASHEFF 19 My time as Mike Boardman's student and our work on infinite loop spaces R. M. VOGT 29 Research Papers Stabilizing the lower operations for mod two cohomology T. BISSON, D. PENGELLEY, AND F. WILLIAMS 39 Conditionally convergent spectral sequences J. M. BOARDMAN 49 On K ( n )-equivalences of spaces A. K. BOUSFIELD 85 Tesselations of moduli spaces and the mosaic operad S. DEVADOSS 91 Hop£ rings, Dieudonne modules, and ES22 8 3 P. G. GOERSS 115 The structure of the Bousfield lattice M. HOVEY AND J. H. PALMIERI 175 vii viii CONTENTS Transfinite spectral sequences Po Hu 197 The Z/p-equivariant complex cobordism ring l. KRIZ 217 Hop£ constructions, Samelson products, and suspension maps K. MORISUGI 225 Embedding homotopy spheres and the Kervaire invariant D. RANDALL 239 Adjoining roots of unity to E 00 ring spectra in good cases -a remark R. SCHWANZL, R. M. VOGT, AND F. WALDHAUSEN 245 Grothendieck topology and the Picard group of a complex orbifold B. STEER AND A. WREN 251 Formal schemes and formal groups N. P. STRICKLAND 263 Simplicial commutative Fp-algebras through the looking-glass of the Fp-local spaces J. M. TURNER 353 The swiss-cheese operad A. A. VORONOV 365 K(n + 1) equivalence implies K(n) equivalence W. S. WILSON 375 Preface The 1998 AMS Winter Meeting in Baltimore was the occasion for a series of special sessions in homotopy theory honoring our colleague J. Michael Boardman on his 60th birthday, and this volume has grown from that beginning. We hope it reflects the influence of his thinking and his personality. Mike is famous for his reticence and his wit. Having created modern stable homotopy theory with his Warwick notes, he has been content to leave them un- published. We are proud to have used this volume as the occasion to bring his classic work on conditionally convergent spectral sequences into print. His thesis on the foundations of Thorn's theory of singularities is a cornerstone of that subject, and much of modern differential topology and K-theory rest on the Boardman-Vogt theory of homotopy-everything H -spaces. One of our goals in this volume has been to show how the trees Mike planted have grown. More than fifty people spoke in the series of special sessions, but one afternoon was explicitly devoted to capturing some of the history of algebraic topology in the 60's and 70's; this volume begins with those accounts. The bulk of this festschrift, however, consists of research papers, which we hope will give some sense of the vitality of the theory of 'homotopy-invariant algebraic structures'. It was on Mike's watch that homotopy theory became autonomous: the notion that geometric objects can be modelled by algebraic structures goes back to the Renaissance, but at some point in the sixties people began to sense the possibility of thinking of spaces themselves as algebraic objects. Frank Adams demonstrated very convincingly the power of homological algebra in stable categories, and Quillen showed that many problems previously regarded as pure algebra were in fact pro- foundly homotopy-theoretic in nature. Much of Mike's work has been concerned with developing the quite subtle ways of thinking needed to do algebra in homo- topy categories; one line of descent leads from that work to Waldhausen's brave new rings, but others lead to other applications, from operads to logic. It is our pleasure and our good fortune to be Mike Boardman's colleagues. We present this volume in his honor and hope that it reflects the respect of his many friends who attended the conference or submitted papers. Jean-Pierre Meyer Jack Morava W. Stephen Wilson ix References-Publications by J. Michael Boardman [1] Some embeddings of 2-spheres in 4-manifolds, Proc.
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