Cohomology Operations and Applications to Homotopy Theory

Mosher R.E., Tangora M.C.

Cohomology operations are at the center of a major area of activity in algebraic topology. This treatment explores the single most important variety of operations, the Steenrod squares. It constructs these operations, proves their major properties, and provides numerous applications, including several different techniques of homotopy theory useful for computation.

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Table of contents :
Cohomology Operations and Applications in Homotopy Theory……Page 1
Contents……Page 5
Preface……Page 9
1. Introduction to cohomology operations……Page 11
Cohomology operations and K(π,n) spaces……Page 12
The machinery of obstruction theory……Page 16
Applications of obstruction theory……Page 17
Discussion……Page 20
References……Page 21
The complex K(Z2,1)……Page 22
The acyclic carrier theorem……Page 24
Construction of the cup-i products……Page 25
The squaring operations……Page 26
Compatibility with coboundary and suspension……Page 28
Exercise……Page 30
References……Page 31
3. Properties of the squares……Page 32
Sq1 and Sq0……Page 33
The Cartan formula and the homomorphism Sq……Page 34
Squares in the n-fold Cartesian product of K(Z2,1)……Page 36
The Adem relations……Page 39
References……Page 42
Properties of the Hopf invariant……Page 43
Decomposable operations……Page 46
Discussion……Page 47
References……Page 48
k-fields and V_{n,k+1}……Page 49
A cell decomposition of V_{n,k}……Page 50
The cohomology of P_{n,k}……Page 53
References……Page 54
Graded modules and algebras……Page 55
The Steenrod algebra A……Page 56
The diagonal map of A……Page 57
Hopf algebras……Page 59
The dual of the Steenrod algebra……Page 60
Algebras over a Hopf algebra……Page 62
The diagonal map of A*……Page 63
The Milnor basis for A……Page 66
Discussion……Page 67
References……Page 68
Exact couples……Page 69
The Bockstein exact couple……Page 70
The spectral sequence of a filtered complex……Page 72
Example: the homology of a cell complex……Page 77
Double complexes……Page 78
Appendix: the homotopy exact couple……Page 80
Exercises……Page 81
References……Page 82
Fibre spaces……Page 83
Example: H*(ΩSn)……Page 86
Serre’s exact sequence……Page 87
The cohomology spectral sequence of a fibre space……Page 90
Discussion……Page 91
References……Page 92
Two types of fibre spaces……Page 93
Calculation of H*(Z2,2;Z2)……Page 96
H*(Z2,q;Z2) and Borel’s theorem……Page 98
Further special cases of H*(π,n;G)……Page 99
Discussion……Page 100
Appendix: proof of Borel’s theorem……Page 101
References……Page 102
Elementary properties of classes……Page 103
Topological theorems mod C……Page 105
The Hurewicz theorem……Page 106
Cp satisfies axiom 3……Page 108
The Cp approximation theorem……Page 110
References……Page 111
Induced fibre spaces……Page 112
The transgression of the fundamental class……Page 113
Bocksteins and the Bockstein lemma……Page 114
Principal fibre spaces……Page 118
Discussion……Page 119
References……Page 120
The suspension theorem……Page 121
A better approximation to Sn……Page 122
Calculation of π_{n+k}(Sn), k ≤ 7……Page 124
The calculation continues……Page 128
Discussion……Page 132
Appendix: some homotopy groups of S3……Page 133
References……Page 137
13. n-type and Postnikov systems……Page 138
n-type……Page 139
Postnikov systems……Page 141
Existence of Postnikov systems……Page 142
Naturality and uniqueness……Page 145
Discussion……Page 146
References……Page 147
The fibre mapping sequence……Page 148
appings of low-dimensional complexes into a sphere……Page 150
The spectral sequence for [K,X]……Page 152
The skeleton filtration and the inclusion mapping sequence……Page 153
Appendix: properties of the fibre mapping sequence……Page 155
The map from ΩE to ΩB……Page 156
References……Page 158
Natural group structures……Page 159
Examples: cohomology and homotopy groups……Page 160
Consequences of Serre’s exact sequence……Page 161
Discussion……Page 164
References……Page 165
Functional cohomology operations……Page 166
Another formulation of θf……Page 169
Secondary cohomology operations……Page 171
Secondary operations and relations……Page 172
The Peterson-Stein formulas……Page 174
Exercises……Page 178
References……Page 179
Secondary compositions……Page 180
The 0-, 1-, and 2-stems……Page 183
The 3-stem……Page 188
The 6- and 7-stems……Page 190
Discussion……Page 192
References……Page 193
18. The Adams spectral sequence……Page 194
Resolutions……Page 195
The Adams filtration……Page 197
Evaluation at F∞……Page 198
The stable groups {Y,X}……Page 201
The definition of Ext_A(M,N)……Page 202
Properties of the spectral sequence……Page 203
Minimal resolutions……Page 209
Some values of Ext^{s,t}_A(Z2,Z2)……Page 211
Applications to homotopy groups……Page 213
References……Page 216
Bibliography……Page 217
Index……Page 222