Stable Homotopy Groups of Spheres A Computer-Assisted Approach
A central problem in algebraic topology is the calculation of the values of the stable homotopy groups of spheres +*S. In this book, a new method for this is developed based upon the analysis of the Atiyah-Hirzebruch spectral sequence. After the tools for
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Stanley O. Kochman
Stable Homotopy Groups of Spheres A Computer-Assisted Approach
Springer-Verlag Berlin Heidelberg NewYork London ParisTokyo Hong Kong
Author
Stanley O. Kochman Department of Mathematics, York University 4700 Keele Street, North York, Ontario M3J 1P3, Canada
Mathematics Subject Classification (1980): Primary: 55045 Secondary: 55T25. 55S30, 55050 ISBN 3-540-52468-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52468-1 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1990 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
PREFACE
This work develops the theoretical basis for an efficient method for the S
inductive calculation of the stable homotopy groups of spheres, n.. the steps of this method are algorithmic and are done by computer. apply this method to compute the first 64 stable stems.
Most of We will
This method is based
upon the analysis of the Atiyah-Hirzebruch spectral sequence: =
H BP n
H.BP and n.BP are well known. h:n.BP oo = n,O
E
®
n
S t
=> n
n+t
BP.
Moreover, the Hurewicz homomorphism
H.BP is a monomorphism.
Therefore, E
hen BP) which is also well known. n
If n
S
oo
n,t
=0
if t
0,
and
is known for t < T then, with
t
the exception of one step, it is algorithmic to deduce the composition series r
r
I mage l d : E
r,T-r-+l
r
E
'O,T
], 2
oS
r- oS
S
T+1, of n . T
The determination of n
S T
from this composition series, the solution of the "additive extension problem", is accomplished using Toda brackets.
A distinctive feature of this method is that all the hard computations are done by computer.
This includes the determination of differentials using
Quillen operations and the computation of Er+ 1 = Kernel [dr:E r N,t N,t
1 / Image [dr:E r Er N-r,t+r-l N+r,t-r+l
r E 1. N,t
On the other hand there are two key steps which require human intervention in S
the computation of each n : T
(1)
the matching of the I ist of "new" elements in degree T+1 which are hit by differentials with the list of "new" elements in degree T+2 on which nonzero differentials originate;
(2)
the solution of the additive extension problems.
IV
Chapter 1 is devoted to the exposition of the background of this computation and to a detailed description of the method we will use.
Even the most
experienced reader should read the exposition of our notation for elements of the stable st
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