Shape Theory An Introduction
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688 Jerzy Dydak Jack Segal
Shape Theory An Introduction
Springer-Verlag Berlin Heidelberg New York 1978
Authors Jerzy Dydak Institute of Mathematics Polish Academy of Sciences ul Sniadeckich 8 Warszawa/Poland
Jack Segal Department of Mathematics University of Washington Seattle, WA 98195/USA
AMS Subject Classifications (1970): 54C56, 54C55, 55 B05
ISBN 3-540-08955-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08955-1 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 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
to
Arlene and Barbara
CONTENTS Chapter I.
Introduction .
Chapter II.
Preliminaries
Chapter III.
§1.
Topology
6
§2.
Homotopy Theory
8
§3.
Category Theory
11
The Shape Category §1. §2.
Chapter IV.
1
20
Definition of the Shape Category Some properties of the shape category and the shape functor .
26
§3.
Representation of shape morphisms
30
§4.
Borsuk's approach to shape theory
32 36
§5. Chapman's Comrlement Theorem General Properties of the Shape Category and the Shape Functor §1.
Chapter
v.
Chapter VI.
Continuity
47
§2.
Inverse Limits in the Shape Category
51
§3.
Fox's Theorem in Shape Theory . . . .
§4.
Shape Properties of Some Decomposition Spaces
53 56
§5. Space of Components . . . . . . . . . . . . . Shape Invariants v
§1.
Cech homology, cohomology, and homotopy pro-groups . .
64
§2.
Movability and n-movability
64
§3.
Deformation Dimension
68
§4.
Shape retracts
74
Algebraic Properties Associated with Shape Theory §1.
The Mittag-Leffler condition and the use of lim 1 .
77 81
+
Chapter VII.
60
§2. Homotopy idempotents Pointed 1-movability §1.
Definition of pointed 1-movable continua and their properties
88
§2.
Representation of pointed 1-movable continua
93
§3.
Pointed 1-movability on curves
98
. . . .
Chapter VIII. Whitehead and Hurewicz Theorems in Shape Theory §1.
Chapter IX.
Preliminary results . . . . .
103
§2.
The Whitehead Theorem in shape theory
106
§3.
The Hurewicz Theorem in shape theory
109
Characterizations and Properties of Pointed ANSR's §1.
Preliminary results . . . . . . . .
111
§2.
Characterizations of pointed ANSR's
114
VI
Chapter X.
§3.
The Union Theorem for ANSR's
116
§4.
ANR-divisors
119
Cell-like Maps §1.
Preliminary definitions and results
123
§2.
The Smale Theorem in shape theory .
127
§3.
Examples of cell-like maps which are not shape equivalences
§4. Chapter XI.
.
. .
.
Hereditary shape equivalences
S
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