Infinite Loop Spaces (AM-90), Volume 90: Hermann Weyl Lectures, The Institute for Advanced Study. (AM-90) 9781400821259
The theory of infinite loop spaces has been the center of much recent activity in algebraic topology. Frank Adams survey
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English Pages 230 [228] Year 1978
PREFACE
TABLE OF CONTENTS
CHAPTER 1. BACKGROUND AND PRELIMINARIES
§1.1. Introduction
§1.2. Loop-spaces
§1.3. Stable homotopy theory
§1.4. Infinite loop spaces
§1.5. Generalized cohomology theories
§1.6. The relation between spectra and generalized cohomology theories
§1.7. The relation between spectra and infinite loop spaces
§1.8. Survey of examples
CHAPTER 2. MACHINERY
§2.1. Introduction
§2.2. Loop-spaces and A∞ spaces in the sense of Stasheff
§2.3. N-fold and infinite loop spaces; En and E∞ spaces
§2.4. Methods
§2.5. Segal's machine
§2.6. Categorical input
§2.7. Ring-theories
CHAPTER 3. LOCALIZATION AND “GROUP COMPLETION”
§3.1. Localization
§3.2. The "plus construction” and "group completion”
CHAPTER 4. TRANSFER
§4.1. Introduction
§4.2. Transfer and structure maps
§4.3. Formal properties of the transfer
CHAPTER 5. THE ADAMS CONJECTURE
§5.1. Discussion of the conjecture
§5.2. Proofs of the conjecture
CHAPTER 6. THE SPECIAL CASE OF K-THEORY SPECTRA; THE THEOREMS OF ADAMS-PRIDDY AND MADSEN-SNAITH-TORNEHAVE
§6.1. Introduction
§6.2. The theorem of Adams and Priddy
§6.3. The theorem of Madsen, Snaith and Tornehave
§6.4. Calculation of [ku, kuΛ] and proof of Proposition 6.3.4
§6.5. The transfer calculation
§6.6. Slight convergence section
CHAPTER 7. THE STATE OF THE ART
§7.1 Survey
REFERENCES
INDEX
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Annals of Mathematics Studies Number 90
INFINITE LOOP SPACES
BY
J. F. ADAMS Hermann Weyl Lectures The Institute for Advanced Study
PRINCETON UNIVERSITY PRESS
PRINCETON, N E W JERSEY 1978
Copyright © 197S by Princeton University Press A ll Rights Reserved Library o f Congress Cataloging-in-Publicaiion Data Adams, John Frank. Infinite loop spaces. (Annals of mathematics studies; no. 90) (Hermann Weyl lectures) Bibliography: p. 1. Loop spaces. 1. Title. II. Series. III. Series: Hermann Weyl lectures. Q A 6 12.76.A3 514\24 78-51152 ISBN 0-691-08207-3 ISBN 0-691-08206-5 pbk. Printed in the United States o f America
HERMANN WEYL LE CTUR ES
The Hermann Weyl Lectures are organized and sponsored by the School of Mathematics of the Institute for Advanced Study.
Their aim is
to provide broad surveys of various topics in mathematics, accessible to nonspecialists, to be published eventually in the Annals of Mathematics Studies. The present monograph is the third in the series.
It is an outgrowth
of the sixth set of Hermann Weyl Lectures, which consisted of six lectures given by Professor J. F. Adams at the Institute for Advanced Study on March 18, 19, 24, 26, April 1, 2, 1975.
ARMAND B O R E L JO H N W. M IL N O R
PREFACE
This book derives from a series of Hermann Weyl Lectures which I gave at the Institute for Advanced Study, Princeton, in the spring of 1975. It is a pleasure to thank my hosts for their invitation, their hospitality, and for providing so discriminating an audience. for my delay in submitting this manuscript.
I should also apologize
In the intervening time some
progress has been made with the theory, and I have taken the opportunity to mention some of it below.
Moreover, a number of other sources have
appeared, and of these [96] and [99] can be recommended as particularly useful to experienced topologists who want to see the results of the sub ject.
However, my object has been a more elementary exposition, which I
hope may convey the basic ideas of the subject in a way as nearly pain less as I can make it.
In this the Princeton audience encouraged me; the
more I found means to omit the technical details, the more they seemed to like it.
If that is the reaction of seasoned topologists, I hope that begin
ners may find it useful to have a gentle introduction to the ideas used in the current literature. I am very grateful to J. P. May, B. J. Sanderson and S. B. Priddy for reading the first draft of this book, in part or in whole; I have benefited greatly from their comments. It goes without saying that I accept the re sponsibility for any jokes which remain.
J. F . A DAM S
TABLE OF CONTENTS C H APTE R 1. BACKGROUND AND PRELIMINARIES §1.1.
Introduction
3
§1.2.
Loop-spaces
3
§1.3.
Stable homotopy theory
§1.4.
Infinite loop spaces
7 12
§1.5. §1.6.
Generalized cohomologytheories The relation between spectra and generalized cohomology theories
§1.7.
The relation between spectra and infinite loop spaces Survey of examples
§1.8.
16 17
22 24
C H APTE R 2. MACHINERY §2.1. Introduction
30
§2.2.
Loop-spaces and spaces in the sense of Stasheff §2.3. N-fold and infinite loop spaces; E n and E ^ spaces §2.4. §2.5.
30 36
Methods SegaVs machine
50 59
§2.6.
Categorical input
65
§2.7.
Ring-theories
70
C H APTE R 3. L O C A LIZ A T IO N AND “ GROUP C O M PLETIO N” §3.1. §3.2.
Localization The (iplus construction” and