A User’s Guide to Spectral Sequences

Cover ......Page 1
Title Page ......Page 2
Preface ......Page 6
Introduction ......Page 8
Table of Contents ......Page 13
Part I: Algebra ......Page 16
1.1. \"There is a spectral sequence ......Page 18
1.2. Lacunary phenomena ......Page 22
1.3. Exploiting further structure ......Page 24
1.4. Working backwards ......Page 34
1.5. Interpreting the answer ......Page 38
2.1. Definitions and basic properties ......Page 43
2.2. How does a spectral sequence arise ......Page 46
2.3. Spectral sequences of algebras ......Page 59
2.4. Algebraic applications ......Page 61
3.1. On convergence ......Page 76
3.2. Limits and colimits ......Page 82
3.3. Zeeman\'s comparison theorem ......Page 97
Part II: Topology ......Page 104
4. Topological Background ......Page 106
4.1. CW-complexes ......Page 107
4.2. Simplicial sets ......Page 118
4.3. Fibrations ......Page 124
4.4. Hopf algebras and the Steenrod algebra ......Page 137
5. The Leray-Serre spectral sequence I ......Page 148
5.1. Construction of the spectral sequence ......Page 151
5.2. Immediate applications ......Page 155
5.3. Appendices ......Page 178
6. The Leray-Serre spectral sequence II ......Page 195
6.1. A proof of theorem 6.1 ......Page 196
6.2. The transgression ......Page 200
6.3. Classifying spaces and characteristic classes ......Page 222
6.4. Other constructions of the spectral sequence ......Page 236
7. The Eilenberg-Moore Spectral Sequence I ......Page 247
7.1. Differential homological algebra ......Page 249
7.2. Bringing in the topology ......Page 263
7.3. The Koszul complex ......Page 272
7.4. The homology of quotient spaces of group actions \'2 ......Page 80
8. The Eilenberg-Moore Spectral Sequence II ......Page 288
8.1. On homogeneous spaces ......Page 289
8.2. Differentials in the Eilenberg-Moore spectral sequence ......Page 312
8.3. Further structure ......Page 328
8bis. Nontrivial Fundamental Groups ......Page 344
8bis .1. Actions of the fundamental group ......Page 345
8bis .2. Homology of groups ......Page 349
8bis .3. Nilpotent spaces and groups ......Page 359
9. The Adams Spectral Sequence ......Page 381
9.1. Motivation: What cohomology sees ......Page 383
9.2. More homological algebra; the functor Ext ......Page 391
9.3. The spectral sequence ......Page 407
9.4. Other geometric applications ......Page 422
9.5. Computations ......Page 430
9.6. Further structure ......Page 445
10. The Bockstein spectral sequence ......Page 470
10.1. The Bockstein spectral sequence ......Page 473
10.2. Other Bockstein spectral sequences ......Page 495
Part III: Sins of Omission ......Page 500
11.1. Spectral sequences for mappings and spaces of mappings ......Page 502
11.2. Spectral sequences and spectra ......Page 510
11.3. Other Adams spectral sequences ......Page 514
11.4. Equivariant matters ......Page 516
11.5. Miscellanea ......Page 519
12.1. Spectral sequences for rings and modules ......Page 522
12.2. Spectral sequences in geometry ......Page 530
12.3. Spectral sequences in algebraic K-theory ......Page 535
12.4. Derived categories ......Page 538
Bibliography ......Page 540
Symbol Index ......Page 568
Index ......Page 570