The Algebra of Secondary Cohomology Operations
The algebra of primary cohomology operations computed by the well-known Steenrod algebra is one of the most powerful tools of algebraic topology. This book computes the algebra of secondary cohomology operations which enriches the structure of the Steenro
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Series Editors H. Bass J. Oesterlé A. Weinstein
Hans-Joachim Baues
The Algebra of Secondary Cohomology Operations
Birkhäuser Verlag Basel Boston Berlin
Author: Hans-Joachim Baues Max-Planck-Institut für Mathematik Vivatsgasse 7 53111 Bonn Germany e-mail: [email protected]
2000 Mathematics Subject Classification: Primary 18G10, 55T15, 55S20
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
ISBN 3-7643-7448-9 Birkhäuser Verlag, Basel – Boston – Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF ∞ Printed in Germany ISBN-10: 3-7643-7448-9 e-ISBN: 3-7643-7449-7 ISBN-13: 978-3-7643-7448-8 987654321
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Contents Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
I Secondary Cohomology and Track Calculus 1
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Primary Cohomology Operations 1.1 Unstable algebras . . . . . . . . . . . . . 1.2 Power algebras . . . . . . . . . . . . . . . 1.3 Cartan formula . . . . . . . . . . . . . . . 1.4 Adem relation . . . . . . . . . . . . . . . 1.5 The theory of Eilenberg-MacLane spaces
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31 35 39 43 46 49 52
Calculus of Tracks 3.1 Maps and tracks under and over a space . . . . . . . . 3.2 The partial loop operation . . . . . . . . . . . . . . . . 3.3 The partial loop functor for Eilenberg-MacLane spaces 3.4 Natural systems . . . . . . . . . . . . . . . . . . . . . .
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55 58 63 67
Track Theories and Secondary Cohomology Operations 2.1 The Eilenberg-MacLane spaces Z n . . . . . . . . . . . Appendix to Section 2.1: Small models of Eilenberg-MacLane spaces . . . . . . 2.2 Groupoids of maps . . . . . . . . . . . . . . . . . . . . 2.3 Track categories and track theories . . . . . . . . . . 2.4 Secondary cohomology operations . . . . . . . . . . . 2.5 The secondary Steenrod algebra . . . . . . . . . . . . 2.6 The stable track theory of Eilenberg-MacLane spaces 2.7 Stable secondary cohomology operations . . . . . . .
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