Stable Homotopy Theory
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J. Frank Adams
Stable Homotopy Theory
3 Berkeley, USA 1961
123
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
3 J. Frank Adams Department of Mathematics, University of Manchester
Stable Homotopy Theory Third Edition Lectures delivered at the University of California at Berkeley 1961. Notes by A. T. Vasquez
$ Springer-Verlag Berlin.Heidelberg. New York 1969
National Science Foundation Grant 10 700
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer-Verlag Berlin - Heidelberg 1969 Library of Congress Catalog Card Number 70-90867 • Printed in Germany. Title No. 7323
TABLE OF CONTENTS
I.
Introduction
. . . . . . . . . . . . . . . . .
2.
Prlmary operatlons. Eilenberg-MacLane
(Steenrod Squares, spaces, Milnor's
work on the Steemrod algebra.) . . . . . . . 3.
I
Stabl e homotopy theory.
4
(Construction
and properties of a category of stable objects.) . . . . . . . . . . . . . . 4.
Applications to
of homologica ! algebra
stable homgto~y theory.
(Spectral sequences, etc.) . . . . . . . . . 5.
38
Theorems of perlodiclt~ and approximation
6.
22
in homological algebra . . . .
58
Comments on prospectlve appllcations of 5, work in prosress,
Bibliography
........
etc . . . . . . . . • ..........
Appeudix . . . . . . . . . . . . . . . . . . . . .
69 7~ 75
l)
Introduction Before I get down to the business of exposition,
like to offer a little motivation.
I'd
I want to show that there
are one or two places in homotopy theory where we strongly suspect that there is something systematic ~oing on, but where we are not yet sure what the system is. The first question concerns the stable J-homomorphlsm. I recall that this is a homomorphism j: ~r(S$) ~ ~S (S n) r = ~r+n
n large.
It is of interest to the differential topologists.
Since
Bott, we know that ~r(SO) is periodic with period 8: r = i ~r(SO) = z 2
2
3
4
5
6
7
8
9o,o
0
Z
0
0
0
Z
Z2
Z2...
On the other hand, wSr is not known, but we can nevertheless ask about the behavior of J. prove:
The differential topologists
2
Theorem:
If r = 4k - i, so that ~r(S0) m Z, then J(~r(S0))
Z m where m is a multiple of the denominator of Bk/~k (B~ being in the k th Bernoulli n~nher.) Conjecture:
The above result is best possible,
i.e.
J(~r(S0)) = Z m where m is exactly this denominator. Status of conjecture: ~pnJecture[
No proof in sight.
If r = 8k or 8k + l, so that
~r(SO) = Z2, then J(~r(SO)) = Z 2. Status of conjecture:
Probably provable, but this is
work in progress. The second question is somewhat related to the first; it concerns vector fields on spheres.
We know that S n admits
a continuous field of non-zero tangent vectors if and only if n is odd.
We also know that if n = 1,3,7 then S n is
parallelizable:
that is, S n admits n continuous tangent
vector fields which are linearly independent at every point. The question is then:
for e
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