Cohomology operations and applications in homotopy theory:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Harper & Row
1968
|
| Schriftenreihe: | Harper's series in modern mathematics.
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X,214 S. |
Internformat
MARC
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| 245 | 1 | 0 | |a Cohomology operations and applications in homotopy theory |c Robert E. Mosher ; Martin C. Tangora* |
| 264 | 1 | |a New York |b Harper & Row |c 1968 | |
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Datensatz im Suchindex
| _version_ | 1804121241782583296 |
|---|---|
| adam_text | CONTENTS
PREFACE ix
1 INTRODUCTION TO COHOMOLOGY OPERATIONS 1
Cohomology operations and K(ir,n) spaces 2
The machinery of obstruction theory 6
Applications of obstruction theory 7
2 CONSTRUCTION OF THE STEENROD SQUARES 12
The complex K(Z2,l) 12
The acyclic carrier theorem 14
Construction of the cup-/ products 15
The squaring operations 16
Compatibility with coboundary and suspension 18
3 PROPERTIES OF THE SQUARES 22
Sq1 and Sq° 23
The Cartan formula and the homomorphism Sq 24
Squares in the n-fold Cartesian product of K(Z2, ) 26
The Adem relations 29
4 APPLICATION: THE HOPF INVARIANT 33
Properties of the Hopf invariant 33
Decomposable operations 36
Non-existence of elements of Hopf invariant one 37
v
vi CONTENTS
5 APPLICATION: VECTOR FIELDS ON SPHERES 39
A:-fieldsandKn,*+i 39
A cell decomposition of Kn,t 40
The cohomology ofPa,k 43
6 THE STEENROD ALGEBRA 45
Graded modules and algebras 45
The Steenrod algebra #f 46
Decomposable elements 47
The diagonal map of stf 47
Hopf algebras 49
The dual of the Steenrod algebra 50
Algebras over a Hopf algebra 52
The diagonal map of $4* 53
The Milnor basis for .s/ 56
7 EXACT COUPLES AND SPECTRAL SEQUENCES 59
Exact couples 59
The Bockstein exact couple 60
The spectral sequence of a filtered complex 62
Example: The homology of a cell complex 67
Double complexes 68
Appendix: The homotopy exact couple 70
8 FIBRE SPACES 73
Fibre spaces 73
The homology spectral sequence of a fibre space 76
Example: H*(flS ) 76
Serre s exact sequence 77
The transgression 80
The cohomology spectral sequence of a fibre space 80
9 COHOMOLOGY OF K(n,n) 83
Two types of fibre spaces 83
Calculation of H*(Z2,2; Z2) 86
H*(Z2,q; Z2) and Borel s theorem 88
Further special cases of H*{tt,ii; G) 89
Appendix: Proof of Borel s theorem 91
10 CLASSES OF ABELIAN GROUPS 93
Elementary properties of classes 93
Topological theorems mod t 95
CONTENTS vii
The Hurewicz theorem 96
The relative Hurewicz theorem 98
^p satisfies Axiom 3 98
The ^p approximation theorem 100
11 MORE ABOUT FIBER SPACES 102
Induced fibre spaces 102
The trangression of the fundamental class 103
Bocksteins and the Bockstein lemma 104
Principal fibre spaces 108
12 APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 111
The suspension theorem 111
A better approximation to S 112
Calculation of n,+t(S ), k 7 114
The calculation continues 118
Appendix: Some homotopy groups of S3 123
13 n-TYPE AND POSTNIKOV SYSTEMS 128
Cellular approximation 129
w-type 129
Postnikov systems 131
Existence of Postnikov systems 132
Naturality and uniqueness 135
14 MAPPING SEQUENCES AND HOMOTOPY CLASSIFICATION 138
The fibre mapping sequence 138
Mappings of low-dimensional complexes into a sphere 140
The spectral sequence for [K,X] 142
The skeleton filtration and the inclusion mapping sequence 143
Appendix: Properties of the fibre mapping sequence 145
The map from C1E to Qfl 146
15 PROPERTIES OF THE STABLE RANGE 149
Natural group structures 149
Examples: Cohomology and homotopy groups 150
Consequences of Serre s exact sequence 151
16 HIGHER COHOMOLOGY OPERATIONS 156
Functional cohomology operations 156
Another formulation of 6f 159
Secondary cohomology operations 161
viii CONTENTS
Secondary operations and relations 162
The Peterson-Stein formulas 164
17 COMPOSITIONS IN THE STABLE HOMOTOPY OF SPHERES 170
Secondary compositions 170
The 0-, 1-, and 2-stems 173
The 3-stem 178
The 6- and 7-stems 180
18 THE ADAMS SPECTRAL SEQUENCE 184
Resolutions 185
The Adams filtration 187
Evaluation of F ° 188
The stable groups {Y, X} 191
The definition of Ext*,(M,N) 192
Properties of the spectral sequence 193
Minimal resolutions 199
Some values of Ext jf (Z2,Z2) 201
Applications to homotopy groups 203
BIBLIOGRAPHY 207
INDEX 212
|
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| indexdate | 2024-07-09T16:54:51Z |
| institution | BVB |
| language | English |
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| physical | X,214 S. |
| publishDate | 1968 |
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| publisher | Harper & Row |
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| series2 | Harper's series in modern mathematics. |
| spelling | Mosher, Robert E. Verfasser aut Cohomology operations and applications in homotopy theory Robert E. Mosher ; Martin C. Tangora* New York Harper & Row 1968 X,214 S. txt rdacontent n rdamedia nc rdacarrier Harper's series in modern mathematics. Déformations continues (Mathématiques) Homologie Homology theory Homotopy theory Kohomologie (DE-588)4031700-6 gnd rswk-swf Homotopietheorie (DE-588)4128142-1 gnd rswk-swf Kohomologie (DE-588)4031700-6 s Homotopietheorie (DE-588)4128142-1 s DE-604 Tangora, Martin C. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004489213&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mosher, Robert E. Tangora, Martin C. Cohomology operations and applications in homotopy theory Déformations continues (Mathématiques) Homologie Homology theory Homotopy theory Kohomologie (DE-588)4031700-6 gnd Homotopietheorie (DE-588)4128142-1 gnd |
| subject_GND | (DE-588)4031700-6 (DE-588)4128142-1 |
| title | Cohomology operations and applications in homotopy theory |
| title_auth | Cohomology operations and applications in homotopy theory |
| title_exact_search | Cohomology operations and applications in homotopy theory |
| title_full | Cohomology operations and applications in homotopy theory Robert E. Mosher ; Martin C. Tangora* |
| title_fullStr | Cohomology operations and applications in homotopy theory Robert E. Mosher ; Martin C. Tangora* |
| title_full_unstemmed | Cohomology operations and applications in homotopy theory Robert E. Mosher ; Martin C. Tangora* |
| title_short | Cohomology operations and applications in homotopy theory |
| title_sort | cohomology operations and applications in homotopy theory |
| topic | Déformations continues (Mathématiques) Homologie Homology theory Homotopy theory Kohomologie (DE-588)4031700-6 gnd Homotopietheorie (DE-588)4128142-1 gnd |
| topic_facet | Déformations continues (Mathématiques) Homologie Homology theory Homotopy theory Kohomologie Homotopietheorie |
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