Unstable Homotopy from the Stable Point of View
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368 R. James Milgram Stanford University, Stanford, CAIUSA
Unstable Homotopy from the Stable Point of View
Springer-Verlag Berlin· Heidelberg· New York 1974
AMS Subject Classifications (1970): 55-02, 55E10, 55E20, 55E25, 55E40, 55E99, 55 0 20,55 035, 55 040, 55099 ISBN 3-540-06655-1 Springer-Verlag Berlin· Heidelberg· New York ISBN 0-387-06655-1 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 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.
© by Springer-Verlag Berlin· Heidelberg 1974. Library of Congress Catalog Card Number 73-21213. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
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This work had its origins in two projects.
The first, undertaken
in 1970 with Elmer Rees, was to construct certain low dimensional embeddings of real projective spaces.
In order to do this we needed methods for
calculating unstable homotopy groups of truncated projective spaces and associate spaces, as well as their images under various suspension
homomorphis~s.
Freudenth~
The second was to understand Mahowald's work on
the metastable homotopy of spheres. In 1971 and 1972 my work in surgery made me enlarge the scope of the project and. consider an apparently unrelated problem - the stable homoiDpyof the Eilenberg-MacLane spacesK(Q/Z,n). By means of appropriate fiberings these questions are seen to be merely different faces of the same coin. Hence, this current work which provides a relatively effective fra.me>>Ork for considering such questions.
We generalize Mahowald's
constructions to allow us to apply Adams' spectral sequence techniques to calculations, and we give detailed calculations for many examples; in particular,. those needed for the work with Rees, and. those needed in surgery with coefficients.
CONTENTS
Introduction • • • •
1
§O. Iterated loop spaces
11
§1. The inclusion
26
X + Qnrnx
34 §3. The cohomology o:r the
37
Fn
§4. The structure o:f iterated loop spaces in the metastable range.
44
PART II.
§6. An unstable Adams spectral sequence ••
51
§7. The loop space :functor for resolutions
56
§8. The metastable exact sequence
61
§9. Calculating the groups Exti,j ~( 2 )(H*(F 1 )/A, PART III.
z2 )
70
K(rr,n)'s
75
Applications and Examples
§10. Calculations of the stable homotopy of the
§11. Some calculations of the stable homotopy groups for the K(Z,n).
81
§12. An example for the metastable exact sequence ••
91
§13. Further calculations for some truncated projective spaces
98
References •
107
Introduction In recent years, stable homotopy theory has become a standard tool for the working topologist.
If
~-
the stable homotopy groups of Hi(X,Y)
{Y ,Y , ...
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