The Algebra of Secondary Cohomology Operations (eBook)

Contents 5
Introduction 11
Part I Secondary Cohomology and Track Calculus 33
1 Primary Cohomology Operations 35
1.1 Unstable algebras over the Steenrod algebra 35
1.2 Power algebras 41
1.3 Cartan formula 49
1.4 Adem relation 52
1.5 The theory of Eilenberg-MacLane spaces 54
2 Track Theories and Secondary Cohomology Operations 57
2.1 The Eilenberg-MacLane spaces 57
2.2 Groupoids of maps 67
2.3 Track categories and track theories 71
2.4 Secondary cohomology operations 75
2.5 The secondary Steenrod algebra 78
2.6 The stable track theory of Eilenberg-MacLane spaces 81
2.7 Stable secondary cohomology operations 84
3 Calculus of Tracks 86
3.1 Maps and tracks under and over a space 86
3.2 The partial loop operation 89
3.3 The partial loop functor for Eilenberg-MacLane spaces 94
3.4 Natural systems 98
3.5 Track extensions 101
3.6 Cohomology of categories 103
3.7 Secondary cohomology and the obstruction of Blanc 107
3.8 Secondary cohomology as a stable model 110
4 Stable Linearity Tracks 112
4.1 Weak additive track extensions 112
4.2 Linearity tracks 114
4.3 The G-structure of the secondary Steenrod algebra 124
4.4 The cocycle of [[Kstable p ]] 130
4.5 The Kristensen derivation 137
4.6 Obstruction to linearity of cocycles 145
5 The Algebra of Secondary Cohomology Operations 150
5.1 Track algebras, pair algebras and crossed algebras 150
5.2 The 154
pseudo 154
functor 154
5.3 The stricti.cation of a 161
track 161
algebra 161
5.4 The stricti.cation of a 165
track module 165
5.5 The stricti.cation of the secondary 166
Steenrod algebra 166
5.6 The stricti.cation of secondary cohomology 174
and Kristensen operations 174
5.7 Two-stage operation algebras 177
Part II Products and Power Maps in Secondary Cohomology 180
6 The Algebra Structure of Secondary Cohomology 182
6.1 Permutation algebras 182
6.2 Secondary permutation algebras 189
6.3 Secondary cohomology as a secondary 194
permutation algebra 194
6.4 Induced homotopies 196
6.5 Squaring maps 199
6.6 Secondary cohomology of a product space 202
7 The Borel Construction and Comparison Maps 207
7.1 The Borel construction 207
7.2 Comparison maps 211
7.3 Comparison tracks 214
8 Power Maps and Power Tracks 221
8.1 Power maps 221
8.2 Linearity tracks for power maps 224
8.3 Cartan tracks for power maps 229
8.4 Adem tracks for power maps 232
8.5 Cohomology as a power algebra 235
8.6 Bockstein tracks for power maps 239
9 Secondary Relations for Power Maps 251
9.1 A list of secondary relations 251
9.2 Secondary linearity relations 257
9.3 Relations for iterated linearity tracks 265
9.4 Permutation relations 271
9.5 Secondary Cartan relations 275
9.6 Cartan linearity relation 279
10 Künneth Tracks and Künneth-Steenrod Operations 289
10.1 K ¨ unneth tracks 289
10.2 K ¨ unneth-Steenrod operations 292
10.3 Linearity tracks for K ¨ unneth-Steenrod operations 295
10.4 Cartan tracks for K ¨ unneth-Steenrod operations 298
10.5 The interchange relation for Cartan tracks 304
10.6 The associativity relation for Cartan tracks 306
10.7 The linearity relation for Cartan tracks 310
10.8 Stable K ¨ unneth-Steenrod operations 311
11 The Algebra of A-tracks 319
11.1 The Hopf-algebra 319
11.2 A-tracks 322
11.3 Linearity tracks 324
11.4 Sum and product of A-tracks 325
11.5 The algebra T. of A-tracks 328
11.6 The algebra of linear A-tracks 330
11.7 Generalized Cartan tracks and the associativity relation 332
11.8 Stability of Cartan tracks 337
11.9 The relation diagonal 340
11.10 The right action on the relation diagonal 344
12 Secondary Hopf Algebras 348
12.1 The monoidal category of [p]-algebras 348
12.2 The secondary diagonal 357
12.3 The right action on the secondary diagonal 360
12.4 The secondary Hopf algebra 361
13 The Action of B on Secondary Cohomology 364
13.1 Pair algebras over the secondary Hopf algebra 364
13.2 Secondary cohomology as a pair algebra over 368
13.3 Secondary Instability 376
14 Interchange and the Left Action 378
14.1 The operators 378
and 378
14.2 The extended left action operator 380
14.3 The interchange acting on secondary cohomology 382
14.4 Computation of the extended left action 387
14.5 Computation of the extended symmetry 394
14.6 The track functor 400
15 The Uniqueness of the Secondary Hopf Algebra ß 404
15.1 The A-class of B 404
15.2 Computation of the A-class 411
15.3 The multiplication class of 413
15.4 Proof of the uniqueness theorem 418
15.5 Right equivariant cocycle of 424
16 Computation of the Secondary Hopf Algebra 427
16.1 Right equivariant splitting of 427
16.2 Computation of 432
and the diagonal 432
of 432
16.3 The multiplication in 436
16.4 Computation of the multiplication map 439
16.5 Admissible relations 446
16.6 Computation of 450
Tables 456
Bibliography 502
Index 506