Algebraic K-Theory I. Proceedings of the Conference Held at the Seattle Research Center of Battelle Memorial Institute, August 28 - September 8, 1972.

343 Pages·1973·16.721 MB·English

by  Hyman Bass (editor)

Lecture Notes in
Mathernafics
Edited byA Dold and B. Eckmann
341
Algebraic K-Theory 1-
Higher K-Theories
Proceedings of the Conference held at the Seattle
ResearchCenter of the Battelle Memorial Institute,
Aug. 28-Sept. 8, 1972
Edited byH. Bass
Springer-Verlag
Berlin Heidelberg New York Tokyo
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contlnuatlon on page 339
Lecture Notes in
Mathernafics
Edited byA Dold and B. Eckmann
341
Algebraic K-Theory 1-
Higher K-Theories
Proceedings of the Conference held at the Seattle
ResearchCenter of the Battelle Memorial Institute,
Aug. 28-Sept. 8, 1972
Edited byH. Bass
Springer-Verlag
Berlin Heidelberg New York Tokyo
Editor
HymanBass
DepartmentofMathematics,ColumbiaUniversity
NewYork,N.Y.10027,USA
1stEdition 1973
2ndPrinting 1986
MathematicsSubjectClassification (1970):13015, 14F15,16A54, 18F25
ISBN3-540-06434-6Springer-VerlagBerlinHeidelbergNewYorkTokyo
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Introduction
A conference on algebraic K-theory was held at the Battelle
Seattle Research Center from August 28 to September 8, 1972, with the joint
support of the National Science Foundation and the Battelle Memorial Institute.
The present volume consists mainly of papers presented at, or stimulated by,
that conference, plus some closely related papers by mathematicians who did
not attend the conference but who have kindly consented to publish their work
here. In addition there are several papers devoted to surveys of subjects
treated at the conference, and to the formulation of open research problems.
It was our intention thus to present a reasonably comprehensive documentation
of the current research in algebraic K-theory, and, if possible, to give this
research a greater coherence than it has heretofore enjoyed. It was particularly
to see the latter aim largely achieved already in the course of
preparing these Proceedings.
Algebraic K-theory has two quite different historical roots both
in geometry. The first is concerned with certain topological obstruction
groups, like the Whitehead groups, and the L-groups of surgery theory. Their
computation, which is in principle an algebraic problem about group rings,
is one of the original missions of algebraic K-theory. It remains a rich source
of new problems and ideas, and an excellent proving ground for new techniques.
The second historical source of algebraic K-theory, from which the
subject draws its name, is Grothendieck's proof of the Riemann-Roch theorem,
and the topological K-theory of Atiyah-Hirzebruch, which has the same point
of departure. Starting from the analogy between projective modules and vector
bundles one is led to seek a K-theory for rings analogous to that of
Atiyah-Hirzebruch for spaces. This enterprise made, at first, only very
limited progress. In the few years preceding this conference, however, several
interesting definitions of higher K-groups were proposed; the relations
between them were far from clear.
Meanwhile the detailed study of K and K had revealed some beautiful
l 2
arithmetic phenomena within the classical groups. This contact with algebraic
number theory had become a major impulse in the subject as well as a theme for
IV
conjectures about the significance of the higher K-groups.
More recently there have appeared definitions and potential
applications of higher K-theory in the framework of algebraic geometry.
As this brief account suggests, a large number of mathematicians,
with quite different motivations and technical backgrounds, had become
interested in aspects of algebraic K-theory. It was not altogether apparent
whether the assembling of these efforts under one rubric was litte more than
an accident of nomenclature. In any case it seemed desireable to gather these
mathematicians, some of whom had no other occasion for serious technical
contact, in a congenial and relaxed setting, and to leave much of what would
ensu.e to mathematical and human chemistry. A consensus of those who were
present is that the experiment was enormously successful. Testimony to this
is the fact that many of the important new results in these volumes were pr
in the few months following the conference, growing out of collaborative
efforts and discussions begun there.
One major conclusion of this research is that all of the higher
K-theories which give the "classical" Kn's for n s; 2 coincide. Thus, in
some sense, the subject of higher algebraic K-theory "exLsts'l, an assertion
some had begun to depair of making. Moreover one now has, thanks largely to
the extraordinary work of Quillen, some very effective tools for calculating
higher K-groups in interesting cases.
The papers that follow are somewhat loosely organized under the
headings: I. Higher K-theories; II. "Classical" algebraic K-theory, and
connections with arithmetic; and III. Hermitian K-theories and geometric
applications. Certain papers, as their titles indicate, contain collections
of research problems. The reader should be warned, however, that because of
the vigorous activity the conference, some of the research problems
posed below are in fact resolved elsewhere in these volumes. The editional
effort necessary to eliminate such instances would have cost an excessive
delay in publication.
I am extremely grateful to the following participants who contributed
v
to the preparation of the survey and research problem articles:
S. Bloch, J. Coates, Keith Dennis, S. Gersten, M. Karoubi, M.P. Murthy,
Ted Petrie,. L. Roberts, J. Shaneson, M. Stein, and R. Swan.
On behalf of the participants I express our thanks to the National
Science Foundation and the Battelle Memorial Institute for their generous
financial support. For the splendid facilities and setting of the Battelle
Seattle Research Center, and for the efficient and considerate services of
its staff, the conference participants were uniformly enthusiastic in their
praise and gratitude.
Finally, I wish to thank Kate March of Columbia University
for her invaluable secretarial and administrative assistance
in organizing the conference, and Robert Martin of Columbia
University for his aid in editing these Proceedings.
H. Bass
Paris, April, 1973
LIST OF PARTICIPANTS AND AUTHORS
Dr. Neil Paul Aboff Dr. Spencer J. Bloch
Department of Mathematics Department of Mathematics
Harvard University Fine Hall
2 Divinity Avenue Princeton University
Cambridge, MA 02138 Princeton, NJ 08540
Dr. Yilmaz Akyildiz Dr. Armand Borel
Department of Mathematics Institute for Advanced Study
University of California Princeton University
at Berkeley Princeton, NJ 08540
Berkeley, CA 94704
Dr. Kenneth S. Brown
Dr. Roger Alperin Department of Mathematics
Department of Mathematics Cornell Univers
Rice University Ithaca, NY 04850
6100 Main Street
Houston, TX 77001 Dr. Sylvain cappell
Department of Theoretical
Dr. Donald W. Anderson Mathematics
Department of Mathematics The Weizman Institute of Science
University of California Rehovoth, Israel
at Sdn Diego
San Diego, CA 92037 Mr. Joe Carroll
Department of Mathematics
Dr. David M. Arnold Harvard University
Department of Mathematics 2 Divinity Avenue
New Mexico State University Cambridge, MA 02138
Las Cruces, NM 88001
Dr. A. J. Casson
Dr. Anthony Bak Department of Mathematics
Departement des Mathematigues Trinity College
2-4 rue du Lievre Cambridge, London, England
Geneve, Switzerland
Dr. Stephen U. Chase
Dr. Hyman Bass Department of Mathematics
Department of Mathematics Cornell University
Columbia University White Hall
Broadway and West 116th Street Ithaca, NY 14850
New York, NY 10027
Dr. K. G. Choo
Dr. Israel Berstein Department of Mathematics
Department of Mathematics University of British Columbia
White Hall Vancouver, British Columbia
Carnell University Canada
Ithaca, NY 14850
VIII
Dr. John Henry Coates Dr. Charles H. Giffen
Department of Mathematics Department of Mathematics
Stanford University University of Virginia
Stanford, California 94305 Charlottesville, VA 22904
Dr. Edwin H. Connell Mr. Jimmie N. Graham
Department of Mathematics Department of Mathematics
University of Miami McGill University
Coral Gables, FL 33124 P. O. Box 6070
Montreal 101, Quebec Canada
Dr. Francis X. Connolly
Department of Mathematics Dr. Bruno Harris
University of Notre Dame Department of Mathematics
Notre Dame, IN 46556 Brown University
Providence, RI 02912
Dr. R. Keith Dennis
Department of Mathematics Dr. Allen E. Hatcher
Cornell University Department of Mathematics
White Hall Fine Hall
Ithaca, NY 14850 Princeton University
Princeton, NJ 08540
Dr. Andreas W. M. Dress
Fakultat fur Mathematik Dr. Alex Heller
Universitat Bielefeld, FRG Department of Mathematics
48 Bielefeld Institute for Advanced Study
Postfach 8640, Germany Princeton University
Princeton, NJ 08540
Dr. Richard Elman
Department of Mathematics Dr. Wu-chung Hsiang
Rice University Department of Mathematics
6100 Main Street Fine Hall
Houston, TX 77001 Princeton University
Princeton, NJ 08540
Dr. E. Graham Evans, Jr.
Department of Mathematics Dr. James E. Humphreys
university of Illinois Courant Institute
Urbana, Illinois 61801 New York University
251 Mercer Street
Dr. Howard Garland New York, NY 10012
Department of Mathematics
State University of New York Dr. Dale Husemoller
at Stony Brook Dep3.rtment of Mathematics
Stony Brook, NY 11790 Haverford College
Haverford, PA 19041
Dr. Steve M. Gersten
Department of Mathematics
Rice University
6100 Main Street
Bouston TX 77001
IX
Dr. J. P. Jouanolou Dr. Erhard Luft
Universite de Strasbourg Department of Mathematics
Departement de Mathematique University of British Columbia
7 Rue Rene Descartes Vancouver, British Columbia
67-Strasbourg, France Canada
Dr. Max Karoubi Mr. Robert D. Martin
Faculte des Sciences Room 207, Mathematics Building
Departement de Mathematiques Columbia University
Univesite de Paris VII Broadway and West 116th Street
Quai St.Bernard New York, NY 10027
Parix 5, France
Dr. Serge Maumary
Dr. Stan Klasa Faculte des Sciences
Department of Mathematics Departement des Mathematiques
Carleton university Universite de Lausanue
Ottawa 1, Ontario, Canada Lausanue, Switzerland
Dr. Mark I. Krusemeyer
Dr. Dusa McDuff
Institute for Advanced Study
Department of Pure Mathematics
Princeton, NJ 08540
Cambridge University
16 Mill Lane
Dr. Kee Y. Lam
Cambridge, England
Department of Mathematics
University of British Columbia
Dr. M. Pavarnan Murthy
Vancouver, British Columbia
Department of Mathematics
Canada
university of Chicago
Chicago, IL
Dr. T. Y. Lam
Department of Mathematics
Dr. Richard R. Patterson
University of California
Department of Mathematics
at Berkeley
University of California
Berkeley, CA 94720
at San Diego
La Jolla, CA 92037
Dr. Ronnie Lee
Department of Mathematics
Dr. Claudio Pedrini
Yale University
Instituto di Matematica
New Haven, CT 06520
Via L. B. Alberti 4
16132-Genova, Italy
Dr. Stephen Lichtenbaum
Department of Mathematics
Dr. Ted Petrie
Cornell University Department of Mathematics
Ithaca, NY 14850 Rutgers University
New Brunswick, NJ 08903
Dr. Jean Louis Loday
universite de Strasbourg
Departement de Mathematique
Rue Rene Descartes
67-Strasbourg, France