عملیات همولوژی و کاربردهای آن در نظریه هموتوپی

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. 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