Obstruction Theory on Homotopy Classification of Maps
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628 Hans J. Baues
Obstruction Theory on Homotopy Classification of Maps
Springer-Verlag Berlin Heidelberg New York 1977
Author
Hans J. Baues Sonderforschungsbereich 40 .Jbeoretische Mathematik" Mathematisches Institut der Universitat Wegelerstr. 10 5300 Bonn/BRD
AMS Subject Classification (1970): 55-02, 55A05, 55A20, 55BlO, 55B25, 55B45, 55C25, 55C30, 55DXX, 55EXX, 55GXX, 55H05, 55H15,55H99
ISBN 3-540-08534-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08534-3 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 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210
FUr Charis und Barbara
Contents
CHAPTER O. CONVENTIONS AND NOTATrON (0.0) Maps and homotopy, and excision theorems •.•.•••••• (0.1) Cofibrations and fibrations •.•.•••..••••••.•••••••
15
(0.2) Homotopy groups ••••••••.••••••••••••••••••••••••••
23
(0.3) Whitehead products ••••••••••••••••••••••••••••••••
26
(0.4) Operation of the fundamental group ••••••••••••••••
29
(0.5) Homology and cohomology groups ••.•••••••..••••••••
37
CHAPTER 1. PRINCIPAL FIBRATIONS AND COFIBRATIONS, OBSTRUCTIONS AND DIFFERENCES (1.1) Extension and lifting problems ••••••••••••••••.•.•
35
(1.2) Principal cofibrations and extension of maps and sections. . . . • • • • • . • . . . . . . . . . . . • • • • . . • . . • • . • . • . • • • • .
46
(1.3) Principal fibrations and lifting of maps and retractions •••••••••••••••••••••••••••••••••••••••
68
(1.4) CW-spaces •••••••••••••••••••••••••••••••••••••••••
94
(1.5) Postnikov spaces •••••••.•..•••••.•..•.•...•••••••• 110
CHAPTER 2. RELATIVE PRINCIPAL COFIBRATIONS AND FIBRATIONS (2.1) Relative principal cofibrations ••••••••••••••••••• 121 (2.2) Relative principal fibrations ••••••••••••••••••••• 131 (2.3) Postnikov decompositions, CW-decompositions, and their principal reductions .••.•••.•..•...•.•••••••
141
(2.4) The exact classification sequences of a principal cofibration ••••••••••••.••.•••••••••••••••.••••••• 151 Example: Classification of maps from an n-dimensional Torus into the 2-sphere ••••.•.•...•••••••••
159
VI
(2.5) The exact classification sequences of a principal fibration ••••..•••••••...•.•.••••.••..•.•••••.••••
161
(2.6) The fiber and cofiber sequences in the category of ex-spaces ••••••••••••••••••••••••••••••••••••••
166
CHAPTER 3. ITERATED PRINCIPAL COFIBRATIONS (3.1) The partial suspension ••••••••••••••••••••••••••••
171
Appendix: The algebra of stable and partially stable maps and the Pontrjagin algebra •••..•••••••.•
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