Prospects in Topology

Home , Alejandro Adem, Andrew Casson, Dennis Sullivan, Louis Kauffman, Michael Freedman, Sylvain Cappell

Annals of Mathematics Studies

Number 138

PROCEEDINGS OF A CONFERENCE IN HONOR OF WILLIAM BROWDER

edited by

Frank Quinn

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY

The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein

Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources

Printed in the United States of America by Princeton Academic Press

10 987654321

Library of Congress Cataloging-in-Publication Data

Prospects in topology : proceedings of a conference in honor of W illiam Browder / Edited by Frank Quinn. p. cm. — (Annals of mathematics studies ; no. 138) Conference held Mar. 1994, at Princeton University. Includes bibliographical references. ISB N 0-691-02729-3 (alk. paper). — ISBN 0-691-02728-5 (pbk. : alk. paper) 1. Topology— Congresses. I. Browder, William. II. Quinn, F. (Frank), 1946- . III. Series. QA611.A1P76 1996 514— dc20 95-25751

The publisher would like to acknowledge the editor of this volume for providing the camera-ready copy from which this book was printed PROSPECTS IN TOPOLOGY

F r a n k Q u in n , E d it o r

Proceedings of a conference in honor of William Browder Princeton, March 1994

Contents

Foreword...... vii Program of the conference ...... ix Mathematical descendants of William Browder...... xi A. Adem and R. J. Milgram, The mod 2 cohomology rings of rank 3 simple groups are Cohen-Macaulay...... 3 A. H. Assadi, Algebraic geometric invariants for homotopy actions...... 13 M. Bokstedt and I. Madsen, Algebraic K-theory of local number fields: the un ramified case...... 28 S. E. Cappell and J. L. Shaneson, The mapping cone and cylinder of a stratified map ...... 58 S. Cappell and S. Weinberger, Replacement of fixed sets and of their normal representations in transformation groups of manifolds...... 67 R. Charney and M. W. Davis, Finite K (n ,l)s for Artin groups...... 110 P. J. Eccles, Double point manifolds of immersions of spheres in Euclidean space ...... 125 M. H. Freedman and Z. Wang, Controlled linear algebra...... 138 I. Hambleton and E. K. Pedersen, Non-linear similarity revisited...... 157 J.-C. Hausmann, On the Vietoris-Rips complexes and a cohomology theory for metric spaces...... 175 S. Illman, Every proper smooth action of a Lie group is equivalent to a real analytic action: a contribution to Hilbert’s fifth problem...... 189 G. Katz, Formal deformations of equivariant genera, fixed point formula and universal symmetry blocks...... 221 M. Kreck, W. Liick, and P. Teichner, Stable prime decompositions of four- manifolds ...... 251 J. Morava, Smooth correspondences...... 270

V. Puppe, Simply connected 6-dimensional manifolds with little symmetry and algebras with small tangent space...... 283 vi Contents

F. Quinn, Speculations on Gromov convergence of stratified sets, and Rieman- nian volume collapse...... 303 A. Ranicki, Bordism of automorphisms of manifolds from the algebraic L-theory point of view...... 314 D. Sullivan, Exterior d, the local degree, and smoothability...... 328 Contributors...... 339 Foreword

To the great benefit of Mathematics, William Browder was born in New York City on January 6, 1934. Sixty years later a conference was convened in Princeton to celebrate this event. This volume is the proceedings of that conference. In these sixty years Bill has made remarkable contributions to Mathematics. He has written over n books and research papers. He has made fundamental contributions to three areas: homotopy theory, transformation groups, and the topology of manifolds. He has served as President of the American Mathematical Society, chair of the department at Princeton, editor of the Annals of Mathe matics, and on many influential committees for the Mathematical Society and the National Academy of Sciences. But as remarkable as these achievements are, there is is another which may be even more profound: his influence on younger mathematicians. He has had 28 PhD students, who have in their turn contributed greatly to mathematics. And beyond that he has served as mentor and role model for hundreds of students and young faculty at Princeton and institutions he has visited, and at the Institute for Advanced Study. His pene trating insight, good humor, and evident love of mathematics have inspired an entire generation of mathematicians.

Program of the Conference

S c ie n t if ic P r o g r a m C o m m it e e . Sylvain Cappell, Wu-Chung Hsiang, Frank Quinn (chair), Dennis Sullivan

L o c a l A rrangements C o m m i t t e e . Tony Bahri (chair), Natasha Brunswick, Joe Kohn, Scott Kenney

P r o c e e d in g s . Frank Quinn, editor; Sarah Jaffe, file and copy editor.

L e c t u r e s . Frank Quinn: Welcome, and low-dimensional quantum invariants John Morgan: Applications of Gauge Theory to four-manifold topology Sylvain Cappell: Singularities of analytic functions and counting of lattice points William Pardon: Hodge structure on the L 2-cohomology of an algebraic surface lb Madsen: On the K-theory of complete local rings Shmuel Weinberger: Large Scale Topology and Geometry Soren Illman: Every proper smooth action of a Lie group is equivalent to a real analytic action: A contribution to Hilbert’s fifth problem Andrew Casson: Speculations on the geometrization conjecture Edward Witten: Quantum Gauge Theories In Two And Four Dimensions Amir Assadi: Invariants for Finite Group Actions Julius Shaneson: An Euler-MacLaurin expansion for lattice sums in dimensions above one Dennis Sullivan: A local chart formula for characteristic classes which is alge braic in the *-operator Serguei Novikov: Some applications of hamiltonian foliations of surfaces Andrew Ranicki: Overview of Browder’s work on surgery Alejandro Adem: Overview of Browder’s work on group actions Martin Bendersky: Overview of Browder’s work on homotopy theory Alejandro Adem: Topological K-theory of Arithmetic Groups Louis Kauffman: Knot Theory — What Next? Michael Freedman: Applied Topology

The organizers would like to express their gratitude to the National Science Foundation and the Departments of Mathematics at Princeton University, the University of Wisconsin, and Virginia Tech for partial financial support of the conference.

Mathematical descendants of William Browder

Alejandro Adem Daniel Juan-Pineda Amir Assadi Mingli Chen, Joseph Dolinak II Hamid Egbalnia Semra Ozturk Ergun Yalcin Sylvan Cappell Oliver Attie Ricardo Cruz Amy Davidow William Homer David Miller Washington Mio Stanley Ocken Faiz al-Rubaii Justin Smith Yixin Zhang Rachel Sturm Shmuel Weinberger Steven Curran Min Yan Daniel Chess George Cooke Ken Dahlberg Roy DeMeo Stefano DeMichelis Lanh Dhang Michael Freedman Thomas Kwok Keung Au Stephen H. Brindle Michael Patrick Casey Slava Krushkal Zheng-Xu He Hickling, Fred Huang, Wei Xiao-Song Lin xii Mathematical Descendants of William Browder

(Michael Freedman, cont.) Feng Luo Ping-Zen Ong Zhenghan Wang Shu Yan Paul Green Soren Illman Osmo-Jukka Kanerva Marja Kankaanrinta Vesa Pollanen Jussi Talsi Louis Kaufmann Steven Winker Randall Weiss Norman Levitt Regina Mladineo Santiago Lopez de Medrano George Lusztig R. Bedard C. De Concini I. Grojnowski J. Kelly O. K. Kwon. G. Lawton P. Lees N. O ’Brien J. Smelt N. Spaltenstein, James Maiorana Stavros Papastavridis William Pardon Fernando Fernandez-Carmena David Massey Les Reid Gerald J. Porter Curtis Murley Albert Shar Frank Quinn Ivelina Bobtcheva Masayuki Yamasaki Neal Stoltzfus Mathematical Descendants of William Browder xiii

David Stone Dennis Sullivan Elmar Winkelnkemper Edward Hendricks R. Bruce Williams George Kennedy Thomas Zukowski Nancy Cardim Damjan Kobal Curt McMullen Kevin Pilgrim Jeremy Kahn Yunpin Jiang Waldek Paluba Edson deFaria Adam Epstein Andre Carvalho Jun Hu Jim Gendron John Wagoner Gene Raymond Hall Laurence Robert Taylor Constance Elko Emanouil Magiroupoulos Zarko Bizaca Sandor Howard Strauss Henry Churchill King L. Christine Kinsey Ross Evon Staffeldt Karen Lee Vogtmann Martin Bridson Thomas Brady Randolph Stacey Tuler Robert William Dussault Deirdre Wynne Dobbs Paul Frank Zizza Jeffey Lawrence Mclver Lars Kadison William Geller Leslie Badoian Elmar Winkelnkemper Yuen Fat Wong Alexander Zabrodsky