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Lecture Notes in Mathematics Lecture Notes in Mathematics Edited by A. Dold and 13. Eckmann 551 Algebraic K-Theory Proceedings of the Conference Held at Northwestern University Evanston, January 12-16, 1976 Edited by Michael R. Stein Springer-Verlag Berlin. Heidelberg- New York 19 ? 6 Editor Michael R. Stein Department of Mathematics Northwestern University Evanston, I1. 60201/USA Library of Congress Cataloging in Publication Data Main entry under title: Algebraic K-theory. (Lecture notes in mathematics ; 551) Bibliography: p. Includes index. i. K-theory--Congresses. 2 ~ Homology theory-- Congresses. 3. Rings (Algebra)--Congresses. I. Stein, M~chael R., 1943- II. Series: Lecture notes in mathematics (Berlin) ; 551. QAB,I,q8 no. 551 [QA61~.33] 510'.8s [514'.23] 76-~9894 ISBN AMS Subject Classifications (1970): 13D15, 14C99,14 F15,16A54,18 F25, 18H10, 20C10, 20G05, 20G35, 55 El0, 57A70 ISBN 3-540-07996-3 Springer-Verlag Berlin 9Heidelberg 9New York ISBN 0-38?-0?996-3 Springer-Verlag New York 9Heidelberg 9Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. Introduction A conference on algebraic K-theory, jointly supported by the National Science Foundation and Northwestern University, was held at Northwestern University January 12-16, 1976. These proceedings contain papers presented at that conference, survey articles on certain subspecialities represented at the conference, and related papers, some by mathematicians who did not attend the conference. The diversity of mathematical interests subsumed under the title "algebraic K-theory" is by now well-known; a glance at the contents of this volume will confirm this. To deal with this diversity, a large block of time was left free for participants to organize themselves into seminars on topics of their choice. A list of these seminar talks has been included following the list of lectures given to the full group. I have also listed the names of conference participants and of authors of articles along with their addresses as of January 1976. It seems appropriate to mention here that shortly after the end of this conference, Quillen and Sus]in, working independently, found a positive solution to Serre's problemMhich motivated some of the earliest research in the "classical" period of algebraic K-theory. Quillen's solution is to appear in Inventiones Math. On behalf of the participants, I would like to thank the NSF and Northwestern University's College of Arts and Sciences for their financial support. I would also like to thank Madalyn Kuharick and Georgette Savino of Northwestern's Mathematics Department for their excellent administrative and secretarial help. Michael R. Stein Evanston, July, 1976 Algebraic K-Theory Conference - List of Talks LECTURES Monday, January 12, 1976 Bloch: K-theory of group schemes Roberts: Reducible curves Hatcher: Some new algebraic K-theories Tuesday, January 13, 1976 van der Kallen: Injective stability for K2 Dennis: Algebraic K-theory and Hochschild homology Grayson: + = Q Wednesday, January 14, 1976 Krusemeyer: Serre's problem Wagoner: Continuouscohomology and K-theory Szczarba: K3(Z) Thursday, January 15, 1976 Giffen: Algebraic K2 and K3 invariants of Hermitian forms Loday: Stable homotopy and higher Whitehead groups Bass: Projective modules over infinite groups Friday, January 16, 1976 Quinn: A new surgery obstraction group Pardon: Localization in L-theory Hausmann: Homology spheres in algebraic K-theory SEMINARS Algebraic qeometry and K-theory Bloch: Some examples in the theory of algebraic cycles Murthy: Cancellation theorems for projective modules on affine surfaces Kazhdan: Some strange groups Vl Cohomology of groups Soule: Cohomologyof SL3(~ Brown: Tate cohomology of infinite groups Alperin: Stability for H2(SUn) Fiedorowicz: Homology of classical groups over finite fields Wagoner: Stability for H.(GL.(A)),A a local ring Evens: Chern classes of inBuced representations Ko and K1 Dayton: SK1 (CX) Martin: NK~(/_~),= finitely generated abelian Magurn: SK~ of dihedral groups Kuku: SK~ of orders Wright: K-theory of the category of invertible algebras K2 Dunwoody: K2 of a Euclidean ring Geller:, KGv theory Green: K2 of a division ring van der Kallen: K2 of a regular local noetherian ring of dlmension 2 injects into K2 of its field of fractions Krusemeyer: Kp(F[x,y]) and related computations Strooker: Tile fundamental group of GL n L-theory Ranicki: Algebraic theory of surgery Topology and K-theory Hatcher: The simple homotopy space Wh(X) May: Remarks on Brauer lifting, Frobenius and KO.(Z) Waldhausen: Computation of pseudo-isotopies by non-additive Q construction Browder: Complete intersections, fixed-point free involutions and the Kervaire invariant Giffen: Segal K-theory Browder: K-theory and stable homotopy Edwards: Steenrod homotopy LIST OF PARTICIPANTS AND AUTHORS Professor Roger Alperin Dr. Barry Dayton Department of Mathematics Department of Mathematics Brown University Northeastern Illinois University Providence, Rhode Island 02912 Bryn Mawr at St. Louis Chicago, Illinois 60625 Mr. David F. Anderson Professor R. K. Dennis Department of Mathematics Department of Mathematics University of Chicago Cornell University Chicago, Illinois 60637 Ithaca, New York 14853 Professor Anthony Bak Professor Andreas Dress Fakult~t Fir Mathematik Fakult~t fQr Mathematik Universit~t Bielefeld Universit~t Bielefeld 48 Bielefeld 48 Bielefeld Federal Republic of~ Germany Federal Republic of Germany Professor Hyman Bass Professor M. J. Dunwoody Department of Mathematics Mathematics Division Columbia University University of Sussex New York, New York 10027 Falmer, Brighton BNI 9QF England Professor Spencer B1och Professor David A. Edwards IHES Department of Mathematics 91440 Bures-sur-Yvette State University of New York France .Binghamton, New York 13901 Professor William Browder Professor Helmut Epp Department of Mathematics Department of Mathematics Princeton University De Paul University Princeton, New Jersey 08540 25 East Jackson Boulevard Chicago, Illinois 60604 Professor Kenneth S. Brown Professor Leonard Evens Department of Mathematics Department of Mathematics Cornell University Northwestern University Ithaca, New York 14853 Evanston, Illinois 60201 Ms. Ruth Charney Professor Zbigniew Fiedorowicz Department of Mathematics Department of Mathematics Princeton University University of Michigan Princeton, New Jersey 08540 Ann Arbor, Michigan 48104 viii Dr. Edward Formanek Dr. J. C. Hausmann Department of Mathematics School of Mathematics University of Chicago Institute for Advanced Study Chicago, Illinois 60637 Princeton, New Jersey 08540 Professor Eric Friedlander Dr. Peter T. Johnstone Department of Mathematics Department of Mathematics Northwestern University University of Chicago Evanston, Illinois 60201 Chicago, Illinois 60637 Professor S. Geller Professor Wilberd van der Kallen Department of Mathematics Department of Mathematics Purdue University Northwestern University West Lafayette, Indiana 47907 Evanston, Illinois 60201 Professor C. H. Giffen Dr. To Kambayashi Department of Mathematics Department of Mathematics University of Virginia Northern Illinois University Charlottesville, Virginia 22901 DeKalb, Illinois 60115 Mr. Daniel R. Grayson Professor David Kazhdan MIT Department of Mathematics Room 2-087 Harvard University Cambridge, Massachusetts 02139 Cambridge, Massachusetts 02138 Professor Sherry M. Green Professor Michel Kervaire Department of Mathematics Department of Math6matiques University of Utah Universit~ de Gen~ve Salt Lake City, Utah 84112 2-4 rue du Li~vre Gen~ve 24, Switzerland Professor Bruno Harris Professor Mark Krusemeyer Department of Mathematics Department of Mathematics Brown University Columbia University Providence, Rhode Island 02912 New York, New York I0027 Professor Allen Hatcher Professor A. O. Kuku School of Mathematics Department of Mathematics Institute for Advanced Study University of Ibadan Princeton, New Jersey 08540 Ibadan Nigeria IX Professor R. Lee Mr. Barton Plumstead Department of Mathematics Department of Mathematics Yale University University of Chicago New Haven, Connecticut 06520 Chicago, Illinois 60637 Mr. H. W. Lenstra, Jr. Professor Stewart Priddy Mathematisch Instituut Department of Mathematics Roeterstraat 15 Northwestern University Amsterdam -C Evanston, Illinois 60201 The Netherlands Dr. Jean-Louis Loday Professor Daniel G. Quillen InstitBt de Recherche Department of Mathematics Mathematique Avanc~e Massachusetts Institute of Technology 7, rue R~n~ Descartes Cambridge, Massachusetts 02139 67084 Strasbourg, France Mr. Bruce Magurn Professor Frank Quinn Department of Mathematics Department of Mathematics Northwestern University Yale University Evanston, Illinois 60201 New Haven, Connecticut 06520 Professor Robert Martin Dr. Andrew Ranicki Department of Mathematics - Box I093 Trinity College Hunter College Cambridge 695 Park Avenue England New York, New York lO021 Professor Peter May Dr. Ulf Rehmann Department of Mathematics Fakult~t f~r Mathematik University of Chicago Universit~t
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