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```I  Secondary Cohomology and Track Calculus
1 Primary Cohomology Operations
1.1    Unstable  algebras  .........................             3
1.2    Power algebras ...........    .................           9
1.3    Cartan formula ...................           .......     17
1.4    Adem  relation  ...................         ........     20
1.5    The theory of Eilenberg-MacLane spaces . . . . . . . . . ...  22
2   Track Theories and Secondary Cohomology Operations
2.1    The Eilenberg-MacLane spaces Z  . . . . . . . . . . . . .  . .  25
Appendix to Section 2.1:
Small models of Eilenberg-MacLane spaces . . . . . . . . ...  31
2.2    Groupoids of maps .........................              35
2.3    Track categories and track theories . . . . . . . . . . . . . . .  39
2.4    Secondary cohomology operations . . . . . . . . . . . . . . . . 43
2.5    The secondary Steenrod algebra . . . . . . . . . . . . . . ..  46
2.6    The stable track theory of Eilenberg-MacLane spaces . . . . .  49
2.7    Stable secondary cohomology operations  . . . . . . . . . . .  52
3   Calculus of Tracks
3.1    Maps and tracks under and over a space . . . . . . . . . ...  55
3.2    The partial loop  operation  ....................        58
3.3    The partial loop functor for Eilenberg-MacLane spaces . . . .  63
3.4    Natural systems  ...................          ......     67
3.5    Track  extensions  ..       .   .   . . . . .   .  . . . . . . . . .....  70
3.6    Cohomology of categories ................... .. .        72
3.7    Secondary cohomology and the obstruction of Blanc .. . . . .  76
3.8    Secondary cohomology as a stable model . . . . . . . . . ...  79
4    Stable Linearity Tracks
4.1    Weak additive track extensions . . . . . . . . . . . . . . . ...  81
4.2    Linearity tracks  .................. ......              83
4.3    The r-structure of the secondary Steenrod algebra . . . . ...  93
4.4    The cocycle of [K able. .....  .................         99
4.5    The Kristensen derivation ................... .. .      106
Appendix to Section 4.5:
Computation of F[p] for p odd  ..................       109
4.6    Obstruction to linearity of cocycles. . . . . . . . . . . .  .  114
5   The Algebra of Secondary Cohomology Operations
5.1    Track algebras, pair algebras and crossed algebras ... . . . .  119
5.2    The r-pseudo functor .....  ..................          123
5.3    The strictification of a F-track algebra . . . . . . . . . . . ...  130
5.4    The strictification of a F-track module . . . . . . . . . . . ...  134
5.5    The strictification of the secondary Steenrod algebra.. . . .  135
5.6    The strictification of secondary cohomology
and  Kristensen  operations  ....................       143
5.7    Two-stage operation algebras . . . . . . . . . . . . . . .  .... 146
II Products and Power Maps in Secondary Cohomology
6   The Algebra Structure of Secondary Cohomology
6.1    Permutation algebras ....................... 151
6.2    Secondary permutation algebras . . . . . . . . . . . . . . ...  158
6.3    Secondary cohomology as a secondary permutation algebra .  163
6.4   Induced homotopies    ...................... 165
6.5    Squaring maps ........................... 168
6.6    Secondary cohomology of a product space . . . . . . . ... . .  171
7   The Borel Construction and Comparison Maps
7.1    The Borel construction ................... ... 177
7.2    Comparison  maps  .........................             181
7.3    Comparison tracks ...................          ......   184
8    Power Maps and Power Tracks
8.1    Power maps   ...................              ........      191
8.2    Linearity tracks for power maps . . . . . . . . . . . . . . ...  194
8.3    Cartan tracks for power maps . . . . . . . . . . . . . . . ...  199
8.4    Adem tracks for power maps . . . . . . . . . . . . . . . ...  202
8.5    Cohomology as a power algebra . . . . . . . . . . . . . . . . . 205
8.6    Bockstein tracks for power maps . . . . . . . . . . . .....  209
9    Secondary Relations for Power Maps
9.1    A list of secondary relations . . . . . . . . . . . . . . . . ...  221
9.2    Secondary linearity relations . . . . . . . . . . . . . . . . ...  227
9.3    Relations for iterated linearity tracks . . . . . . . . . . . ...  235
9.4    Permutation  relations  .......................             241
9.5    Secondary Cartan relations . . . . . . . . . . . . . . .....  . 245
9.6    Cartan linearity relation  . . . . . . . . . . . . . . ..... . . 249
10   Kiinneth Tracks and Kiinneth-Steenrod Operations
10.1   Kiinneth  tracks  ..........................               259
10.2   Kiinneth-Steenrod operations . . . . . . . . . . . . . . . .  .  262
10.3   Linearity tracks for Kiinneth-Steenrod operations ...... . ..265
10.4   Cartan tracks for Kiinneth-Steenrod operations  . . . . . ...  268
10.5   The interchange relation for Cartan tracks  . . . . . . ... . . 274
10.6   The associativity relation for Cartan tracks . . . . . . . . ...  276
10.7   The linearity relation for Cartan tracks . . . . . . . . . .  .  280
10.8   Stable Kiinneth-Steenrod operations . . . . . . . . . . . . ...  281
11   The Algebra of A-tracks
11.1   The Hopf-algebra TG(EA) . . . . . . . . . . . . . . ..... . . 289
11.2   A-tracks ...................              .......... 292
11.3   Linearity tracks Fr and F,  . . . . . . . . . . . . . . . . ...  294
11.4   Sum and product of A-tracks . . . . . . . . . . . . . . . .  .  295
11.5   The algebra TA of A-tracks . . . . . . . . . . . . . . .....  . 298
11.6   The algebra of linear A-tracks . . . . . . . . . . . . . . .  .  300
11.7   Generalized Cartan tracks and the associativity relation . . . . 302
11.8   Stability of Cartan tracks . . . . . . . . . . . . . . ..... . . 307
11.9   The relation  diagonal  .......................            310
11.10  The right action on the relation diagonal . . . . . . . . . . . . 314
12   Secondary Hopf Algebras
12.1   The monoidal category of [p]-algebras . . . . . . . . . . . ...  319
12.2   The secondary  diagonal ......................             328
12.3   The right action on the secondary diagonal . . . . . . . . ...  331
12.4   The secondary Hopf algebra B . . . . . . . . . . . . . . . ...  332
13  The Action of B on Secondary Cohomology
13.1   Pair algebras over the secondary Hopf algebra B . . . . . ...  335
13.2   Secondary cohomology as a pair algebra over B . . . . . . ...  339
13.3   Secondary  Instability  .  . .  . .  . . .  . . . .  . . .  . .  . . .  ...  347
14  Interchange and the Left Action
14.1   The operators S  and  L  ......................          349
14.2   The extended left action operator . . . . . . . . . . . . . ...  351
14.3   The interchange acting on secondary cohomology  . . . .  . ..353
14.4   Computation of the extended left action . . . . . . . . . ...  358
14.5   Computation of the extended symmetry . . . . . . . . . ....  365
14.6   The track  functor 7-I*[]  ......................        371
15   The Uniqueness of the Secondary Hopf Algebra B
15.1   The  A-class of B  ..........................            375
15.2   Computation of the A-class . . . . . . . . . . . . . ......  .  382
15.3   The multiplication class of B  . . . . . . . . . . . . . . ...  384
15.4   Proof of the uniqueness theorem . . . . . . . . . . . . . ...  389
15.5   Right equivariant cocycle of  . . . . . . . . . . . . . . . ...  395
16   Computation of the Secondary Hopf Algebra B
16.1   Right equivariant splitting of B . . . . . . . . . . . . . .....  399
16.2   Computation of ý and the diagonal A1 of B . . . . . . . . ...  404
16.3   The multiplication  in  B  .......................       408
16.4   Computation of the multiplication map . . . . . . . . . . ...  411
16.5   Admissible relations . . . . . . . . . . . . . . ..... ......  418
16.6   Computation  of B  .........................             422

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Library of Congress subject headings for this publication: Algebra, Homological, Sequences (Mathematics)Cohomology operations