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 homotopy 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. Cambridge Phil. Soc. 60 (1964), 354-356. [2] Stable homotopy theory, (preprint, Univ. of Warwick, Nov. 1965). [3] Stable homotopy theory, Chapter V, (preprint, Univ. of Warwick, Jan. 1966). [4] Stable homotopy theory, Chapter VI, (preprint, Univ. of Warwick, July 1966). [5] (jointly with B. Steer) Axioms for Hopf invariants, Bull. Amer. Math. Soc. 12 (1966), 992- 994. [6] The principle of signs, L 'Enseignement Math. 12 (1966), 191-194. [7] (jointly with B. Steer) On Hopf invariants, Comment. Math. Helv. 42 (1967), 180-221. [8] Singularities of differentiable maps, Publ. Math. I.H.E.S. 33 (1967), 21-57. [9] On manifolds with involution, Bull. Amer. Math. Soc. 73 (1967), 136-138. [10] (jointly with R. M. Vogt) Homotopy-everything H-spaces, Bull. A mer. Math. Soc. 14 (1968), 1117-1122. [11] Stable homotopy theory, Chapter I, (preprint, Johns Hopkins Univ., Aug. 1969). [12] Stable homotopy theory, Chapter II, (preprint, Johns Hopkins Univ., July 1970). [13] Stable homotopy theory is not self-dual, Proc. Amer. Math. Soc. 26 (1970), 369-370. [14] Stable homotopy theory, Appendix B, (preprint, Johns Hopkins Univ., Nov. 1970). [15] Homotopy structures and the language of trees, Proc. Symp. Pure Math. 22 (1971), 37-58. [16] Cobordism of involutions revisited, Proc. 2nd. Conf. on Compact Transformation Groups, Amherst 1971, Springer Lecture Notes in Math. 298 (1972), 131-151. [17] (jointly with R. M. Vogt) Homotopy invariant algebraic structures on topological spaces, Springer Lecture Notes in Math. 341 (1973). [18] Stable homotopy theory, Appendix C- Localization theory, (preprint, Johns Hopkins Univ., Aug. 1975). [19] Stable homotopy theory, Appendix D - Localization and splittings of MU, (preprint, Johns Hopkins Univ., Feb. 1976). [20] Splittings of MU and other spectra, Springer Lecture Notes in Math. 658 (1978), 27-79. [21] (jointly with R. M. Vogt) Tensor products of theories, application to loop spaces, J. Pure Appl. Alg. 14 (1979), 117-129. [22] (jointly with R. M. Vogt) Gomotopicheski invariantnie algebraicheskie strukturi na topo- logicheskikh prostranstvakh, Izdat. Mir (Moscow, 1977). (Russian translation of [17]) [23] Original Brown-Peterson spectra, Springer Lecture Notes in Math. 741 (1979), 355-372. [24] Graded Eilenberg-MacLane ring spectra, Amer. J. Math. 102 (1980), 979-1010. [25] Conditionally convergent spectral sequences, (preprint, Johns Hopkins Univ., Aug. 1981). [26] The eightfold way to BP-operations, Canad. Math. Soc. Conf. Proc. 2 (1982),. 187-226. [27] Another basis of QBP.BP., (preprint, Johns Hopkins Univ., July 1982). [28] Two more bases of QBP.BP., (preprint, Johns Hopkins Univ., June 1984). [29] Stable and unstable objects for BP-cohomology, (preprint, Johns Hopkins Univ., Mar. 1986). [30] The eightfold way in ordinary homology and cohomology, (preprint, Johns Hopkins Univ., May 1986). [31] Operations on the Adams spectral sequences for Brown-Peterson homology and cohomology, (preprint, Johns Hopkins Univ., Nov. 1988). [32] Modular representations on the homology of powers of real projective space, Contemp. Math. 146 (1993), 49-70. [33] Stable operations in generalized cohomology, Chapter 14 in Handbook of Algebraic Topology (ed. I. M. James), Elsevier (Amsterdam, 1995), 585-686. [34] (jointly with D. C. Johnson, W. S. Wilson) Unstable operations in generalized cohomology, Chapter 15 in Handbook of Algebraic Topology (ed. I. M. James), Elsevier (Amsterdam, 1995), 687-828. [35] (jointly with W. S. Wilson) k(n)-torsion-free H-spaces and P(n)-cohomology, (in prepara- tion). [36] Conditionally convergent spectral sequences, these Proceedings.

xi Selected Titles in This Series (Continued from the front of this publication)

212 E. Ramirez de Arellano, N. Salinas, M. V. Shapiro, and N. L. Vasilevski, Editors, Operator theory for complex and hypercomplex analysis, 1998 211 Jozef Dodziuk and Linda Keen, Editors, Lipa's legacy: Proceedings from the Bers Colloquium, 1997 210 V. Kumar Murty and Michel Waldschmidt, Editors, Number theory, 1998 209 Steven Cox and Irena Lasiecka, Editors, Optimization methods in partial differential equations, 1997 208 MichelL. Lapidus, Lawrence H. Harper, and Adolfo J. Rumbos, Editors, Harmonic analysis and nonlinear differential equations: A volume in honor of Victor L. Shapiro, 1997 207 Yujiro Kawamata and Vyacheslav V. Shokurov, Editors, Birational algebraic geometry: A conference on algebraic geometry in memory of Wei-Liang Chow (1911-1995), 1997 206 Adam Koranyi, Editor, Harmonic functions on trees and buildings, 1997 205 Paulo D. Cordaro and Howard Jacobowitz, Editors, Multidimensional complex analysis and partial differential equations: A collection of papers in honor of Franc;ois Treves, 1997 204 Yair Censor and Simeon Reich, Editors, Recent developments in optimization theory and nonlinear analysis, 1997 203 Hanna Nencka and Jean-Pierre Bourguignon, Editors, Geometry and nature: In memory of W. K. Clifford, 1997 202 Jean-Louis Loday, James D. Stasheff, and Alexander A. Voronov, Editors, Operads: Proceedings of Renaissance Conferences, 1997 201 J. R. Quine and Peter Sarnak, Editors, Extremal Riemann surfaces, 1997 200 F. Dias, J.-M. Ghidaglia, and J.-C. Saut, Editors, Mathematical problems in the theory of water waves, 1996 199 G. Banaszak, W. Gajda, and P. Krason, Editors, Algebraic K-theory, 1996 198 Donald G. Saari and Zhihong Xia, Editors, Hamiltonian dynamics and celestial mechanics, 1996 197 J. E. Bonin, J. G. Oxley, and B. Servatius, Editors, Matroid theory, 1996 ' 196 David Bao, Shiing-shen Chern, and Zhongmin Shen, Editors, Finsler geometry, 1996 195 Warren Dicks and Enric Ventura, The group fixed by a family of injective endomorphisms of a free group, 1996 194 Seok-Jin Kang, Myung-Hwan Kim, and Insok Lee, Editors, Lie algebras and their representations, 1996 193 Chongying Dong and Geoffrey Mason, Editors, Moonshine, the Monster, and related topics, 1996 192 Tomek Bartoszynski and Marion Scheepers, Editors, Set theory, 1995 191 Tuong Ton-That, Kenneth I. Gross, Donald St. P. Richards, and Paul J. Sally, Jr., Editors, Representation theory and harmonic analysis, 1995 190 Mourad E. H. Ismail, M. Zuhair Nashed, Ahmed I. Zayed, and Ahmed F. Ghaleb, Editors, Mathematical analysis, wavelets, and signal processing, 1995 189 S. A.M. Marcantognini, G. A. Mendoza, M.D. Moran, A. Octavio, and W. 0. Urbina, Editors, Harmonic analysis and operator theory, 1995 188 Alejandro Adem, R. James Milgram, and Douglas C. Ravenel, Editors, Homotopy theory and its applications, 1995

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej. Homotopy Invariant Algebraic Structures A Conference in Honor of J, Michael Boardman Jean-Pierre Meyer, Jack Morava, and W. Stephen Wilson, Editors

This volume presents the proceedings of the conference held in honor of J. Michael Boardman's 60th birthday. It brings into print his classic work on conditionally convergent spectral sequences. Over the past 30 years, it has become evident that some of the deepest questions in algebra are best understood against the background of homotopy theory. Boardman and Vogt's theory of homotopy-theoretic algebraic structures and the theory of spectra, for example, were two benchmark breakthroughs underlying the development of algebraic K-theory and the recent advances in the theory of motives. The volume begins with short note& by Mac Lane, May, Stasheff, and others on the early and recent history of the subject. But the bulk of the volume consists of research papers on topics that have been strongly influenced by Boardman's work. Articles give readers a vivid sense of the current state of the theory of "homotopy-invariant algebraic structures". Also included are two major foundational papers by Goerss and Strickland on applications of methods of algebra (i.e., Dieudonne modules and formal schemes) to problems of topology. Boardman is known for the depth and wit of his ideas. This volume is intended to reflect and to celebrate those fine characteristics.

ISBN 0-8218-1057-X

9 780821 810576 CONM/239