Theory of retracts

S. T Hu

Table of contents :
Title……Page 3
Date-line……Page 4
CONTENTS ……Page 5
PREFACE ……Page 9
1. Retract ……Page 11
2. Adjunction spaces ……Page 13
3. Relation with the extension problem ……Page 16
4. Indempotent maps ……Page 17
5. Closedness ……Page 18
6. Fixed point property ……Page 20
7. Homology properties ……Page 21
8. Cohomology properties ……Page 23
9. Global and local contractibility ……Page 25
10. Local connectedness ……Page 26
11. Deformation retracts ……Page 29
12. Neighborhood retracts ……Page 30
1. Classes of spaces ……Page 33
2. AE and ANE for a class of spaces ……Page 34
3. TietzeTs extension theorem ……Page 37
4. Product of extensors ……Page 39
5. Retract of extensors ……Page 40
6. Open subspaces of ANETs ……Page 42
7. Contractible ANETs ……Page 43
8. Union of open extensors ……Page 44
9. Closed subspaces of extensors ……Page 47
10. Union of closed extensors ……Page 49
11. Canonical coverings ……Page 51
12. Replacement by polytopes ……Page 53
13. Topological linear spaces ……Page 55
14. Dugundji’s extension theorem ……Page 57
15. Unit sphere in normed linear space ……Page 60
16. Metrizable extensors ……Page 62
17. Local ANETs ……Page 68
18. Several lemmas ……Page 69
19. Proof of 17.1 ……Page 73
1. AR and ANR for a class of spaces ……Page 80
2. Eilenberg-Wojdyslawski theorem ……Page 81
3. Relation with extensors ……Page 83
4. Metrizable retracts ……Page 86
5. Special Metrizable retracts ……Page 93
6. AR and ANR ……Page 95
7. Elementary properties ……Page 96
8. Local characterization ……Page 98
9. Simplicial polytopes ……Page 99
10. Polytopes with Whitehead topology ……Page 101
11. Polytopes with metric topology ……Page 105
1. Near maps and small homotopies ……Page 110
2. Homotopy extension property ……Page 116
3. Closed ANR subspaces in an ANR ……Page 119
4. Partial realization of polytopes ……Page 122
5. Small deformations ……Page 131
6. Dominating polytopes ……Page 137
7. Homology groups ……Page 140
8. Bridges and Bridge maps ……Page 145
1. Preliminary definitions ……Page 149
2. Characterizing $LC^n$ by neighborhood extension ……Page 150
3. Characterizing $LC^n$ by neighborhood retraction ……Page 155
4. Characterizing $LC^n$ by partial realization ……Page 156
5. Characterizing $LC^n$ by small homotopy ……Page 159
6. Characterizing $LC^n$ by factorization of maps ……Page 164
7. Finite-dimensional ANRTs ……Page 168
8. $LC^*$ compactum but not ANR ……Page 169
9. $LC^infty$ compactum which is not $LC^*$ ……Page 172
10. Spaces which are $C^n$ and $LC^n$ ……Page 174
11. Finite-dimensional ARTs ……Page 175
1. Adjunction spaces of AR’s and ANRTs ……Page 177
2. Mapping spaces of ARTs and ANR’s ……Page 182
3. Relative mapping spaces ……Page 187
4. Compact AR’s in Euclidean spaces ……Page 190
5. Compact ANR’s in Euclidean spaces ……Page 192
1. Deformation and retraction ……Page 198
2. Equivalence theorem ……Page 199
3. Characterization by deformation ……Page 201
4. Obstructions to deformation ……Page 202
5. Cech cohomology groups ……Page 207
6. Relative bridge maps ……Page 208
7. Maps from $(X, X_0)$ to $(Y, Y_0)$ ……Page 211
8. Necessary and sufficient conditions ……Page 216
BIBLIOGRAPHY ……Page 220