Homotopy Theory

Name: Homotopy Theory
Brand: Academic Press
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Academic Press

Published on 1. January 1959

346 pages

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Front Cover
Homotopy Theory, Volume 8
Copyright Page
Contents
PREFACE
LIST OF SPECIAL SYMBOLS AND ABBREVIATIONS
CHAPTER I. MAIN PROBLEM AND PRELIMINARY NOTIONS
1. Introduction
2. The extension problem
3. The method of algebraic topology
4. The retraction problem
5. Combined maps
6. Topological identification
7. The adjunction space
8. Homotopy problem and classification problem
9. The homotopy extension property
10. Relative homotopy
11. Homotopy equivalences
12. The mapping cylinder
13. A generalization of the extension problem
14. The partial mapping cylinder
15. The deformation problem
16. The lifting problem
17. The most general problem
Exercises
CHAPTER II. SOME SPECIAL CASES OF THE MAIN PROBLEMS
1. Introduction
2. The exponential map p: R S1
3. Classification of the maps S1 S1
4. The fundamental group
5. Simply connected spaces
6. Relation between p1(X, x0) and H1 ( X )
7. The Bruschlinsky group
8. The Hopf theorems
9. The Hurewicz theorem
Exercises
CHAPTER III. FIBER SPACES
1. Introduction
2. Covering homotopy property
3. Definition of fiber space
4. Bundle spaces
5. Hopf fiberings of spheres
6. Algebraically trivial maps X S2
7. Liftings and cross-sections
8. Fiber maps and induced fiber spaces
9. Mapping spaces
10. The spaces of paths
11. The space of loops
12. The path lifting property
13. The fibering theorem for mapping spaces
14. The induced maps in mapping spaces
15. Fiberings with discrete fibers
16. Covering spaces
17. Construction of covering spaces
Exercises
CHAPTER IV. HOMOTOPY GROUPS
1. Introduction
2. Absolute homotopy groups
3. Relative homotopy groups
4. The boundary operator
5. Induced transformations
6. The algebraic properties
7. The exactness property
8. The homotopy property
9. The fibering property
10. The triviality property
11. Homotopy systems
12. The uniqueness theorem
13. The group structures
14. The role of the basic point
15. Local system of groups
16. n-Simple spaces
Exercises
CHAPTER V. THE CALCULATION OF HOMOTOPY GROUPS
1. Introduction
2. Homotopy groups of the product of two spaces
3. The one-point union of two spaces
4. The natural homomorphisms from homotopy groups to homology groups
5. Direct sum theorems
6. Homotopy groups of fiber spaces
7. Homotopy groups of covering spaces
8. The n-connective fiberings
9. The homotopy sequence of a triple
10. The homotopy groups of a triad
11. Freudenthal's suspension
Exercises
CHAPTER VI. OBSTRUCTION THEORY
1. Introduction
2. The extension index
3. The obstruction cn+1 (g)
4. The difference cochain
5. Eilenberg's extension theorem
6. The obstruction sets for extension
7. The homotopy problem
8. The obstruction dn(f, g
ht)
9. The group Rn(K,L
f)
10. The obstruction sets for homotopy
11. The general homotopy theorem
12. The classification problem
13. The primary obstructions
14. Primary extension theorems
15. Primary homotopy theorems
16. Primary classification theorems
17. The characteristic element of Y
Exercises
CHAPTER VII. COHOMOTOPY GROUPS
1. Introduction
2. The cohomotopy set pm( X, A )
3. The induced transformations
4. The coboundary operator
5. The group operation in pm( X, A )
6. The cohomotopy sequence of a triple
7. An important lemma
8. The statement (6)
9. The statement (5)
10. Higher cohomotopy groups
11. Relations with cohomology groups
12. Relations with homotopy groups
Exercises
CHAPTER VIII. EXACT COUPLES AND SPECTRAL SEQUENCES
1. Introduction
2. Differential groups
3. Graded and bigraded groups
4. Exact couples
5. Bigraded exact couples
6. Regular couples
7. The graded groups R( G ) and S( G )
8. The fundamental exact sequence
9. Mappings of exact couples
10. Filtered differential groups
11. Filtered graded differential groups
12. Mappings of filtered graded d-groups
Exercises
CHAPTER IX. THE SPECTRALSEQUENCE OF A FIBER SPACE
1. Introduction
2. Cubical singular homology theory
3. A filtration in the group of singular chains in a fiber space
4. The associated exact couple
5. The derived couple
6. Homology with arbitrary coefficients
7. The spectral homology sequence
8. Proof of Lemma A
9. Proof of Lemma B
10. Proof of Lemmas C and D
11. The Poincaré polynomials
12. Gysin's exact sequences
13. Wang's exact sequences .
14. Truncated exact sequences
15. The spectral sequence of a regular covering space
16. A theorem of P. A. Smith
17. Influence of the fundamental group on homology and cohomology groups
18. Finite groups operating freely on Sr
Exercises
CHAPTER X. CLASSES OF ABELIAN GROUPS
1. Introduction
2. The definition of classes
3. The primary components of abelian groups
4. The G-notions on abelian groups
5. Perfectness and completeness
6. Applications of classes to fiber spaces
7. Applications to n-connective fiber spaces
8. The generalized Hurewicz theorem
9. The relative Hurewicz theorem
10. The Whitehead theorem
Exercises
CHAPTER XI. HOMOTOPY GROUPS OF SPHERES
1. Introduction
2. The suspension theorem
3. The canonical map
4. Wang's isomorphism p*
5. Relation between p* and i#
6. The triad homotopy groups
7. Finiteness of higher homotopy groups of odd-dimensional spheres
8. The iterated suspension
9. The p-primary components of pm(S3)
10. Pseudo-projective spaces
11. Stiefel manifolds
12. Finiteness of higher homotopy groups of even-dimensional spheres
13. The p-primary components of homotopy groups of even- dimensional spheres
14. The Hopf invariant .
15. The groups pn+1(Sn) and pn+2(Sn)
16. The groups pn+3(Sn)
17. The groups pn+4(Sn)
18.The groups pn+r(Sn), 5 = Y = 15
Exercises
BIBLIOGRAPHY
INDEX

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