H-spaces from a Homotopy Point of View
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James Stasheff The Institute for Aqvanced Study Princeton I NJ I USA
H-Spaces from a Homotopy Point of View
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Springer-Verlag Berlin· Heidelberg· NewYork 1970
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 usc, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. C by Springer-Verlag Berlin' Heidelberg 1970. Library of Congress Catalog Card Number Title No. 3318
Printed in Germany.
DEDICATION
In fond memory of George Yuri Rainich who first revealed to me the deep significance yet optional nature of associativity
Preface
These notes have their origin in a course given at Princeton University in the fall of 1968; I am most grateful to Princeton for providing the opportunity to give such a course and to those institutions which provided support during the preparation of these notes: Princeton, the University of Notre Dame, The Institute for Advanced Study, the Alfred P. Sloan Foundation and NSF Grant GP-9590.
The course itself was presaged by a lecture at the Michigan State
Conference on the Topology of Manifolds
The present arrangement
of topics owes much to a short course given at Boston College in the fall of 1969.
Finally, my deep gratitude to Mrs. Ann Gosling of Princeton University
and Miss Evelyn Laur-ent of The Institute of Advanced Study for their fine preparation of preliminary and final typescripts of these notes. An attempt has been made to bring these notes up to date, but current activity is such that the date in question is at best the early spring of 1970. It is hoped that more recent results will be covered at the Conference on H-spaces to be held at the University of Neuchatel in August, 1970, for which these notes should provide adequate background. Bibliographic references are given in the form [Hopf].
Where a given
author has more than one entry in the bibliography, the various entries are distinguished by underlinings, e. g., [Hopf] vs , [!!opf].
Princeton, New Jersey and Lansdale, Pennsylvania
Spring, 1970
TABLE OF CONTENTS Chaeter
PREFACE. • • • . • • • • • • • . . • • • • • • • • • • . . • • • . . • • • • • • • • • • • • • . • •
V
INTRODUC TION •••.••.••••••••.••••..•..•••••••.•••.•..•• 1
THE HOPF CONSTRUCTION.. •• • • •••• ••••••••••••••• ••• ••••
3
2
THE PROJECTIVE PLANE. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
7
3
MAPS INTO AN H-SPACE: ALGEBRAIC STRUCTURE, INVERSES, OTHER MULTIPLICATIONS, ETC.
••••••••••••••• 10
LOOP SPACES AND TOPOLOGICAL GROUPS •• 14
4
ASSOCIATIVITY:
5
H-SPACES WHICH ARE FINITE COMPLEXES. ••••••••••••••••• 20
6
THE BAR CONSTRUCTION SPECTRAL SEQUENCE. • • • • • • • • • • •• 23
7
HOMOTOPY ASSOCIATIVITY •••••••••••••••••••••••••
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