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A USER'S GUIDE TO
SPECTRAL SEQUENCES
Second Edition
JOHN McCLEARY
Vassar College
CAMBRIDGE
UNIVERSITY PRESS
Table of Contents
Preface
Introduction
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vii
ix
Part I: Algebra
1. An Informal Introduction
1.1. "There is a spectral sequence . . . "
1.2. Lacunary phenomena
1.3. Exploiting further structure
1.4. Working backwards
1.5. Interpreting the answer
1
3
3
7
9
19
23
2. What is a Spectral Sequence?
2.1. Definitions and basic properties
2.2. How does a spectral sequence arise?
2.3. Spectral sequences of algebras
2.4. Algebraic applications
28
28
31
44
46
3. Convergence of Spectral Sequences
3.1. On convergence
3.2. Limits and colimits
3.3. Zeeman's comparison theorem
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x
61
61
67
82
Part II: Topology
4. Topological Background
4.1. CW-complexes
4.2. Simplicial sets
4.3. Fibrations
4.4. Hopf algebras and the Steenrod algebra
89
91
92
103
109
122
5. The Leray-Serre spectral sequence I
5.1. Construction of the spectral sequence
5.2. Immediate applications
5.3. Appendices
133
136
140
163
6. The Leray-Serre spectral sequence II
6.1. A proof of theorem 6.1
6.2. The transgression
6.3. Classifying spaces and characteristic classes
6.4. Other constructions of the spectral sequence ®
180
181
185
207
221
7. The Eilenberg-Moore Spectral Sequence I
7.1. Differential homological algebra
7.2. Bringing in the topology
232
234
248
Table of Contents
xv
7.3. The Koszul complex
7.4. The homology of quotient spaces of group actions
257
265
8. The Eilenberg-Moore Spectral Sequence II
8.1. On homogeneous spaces
8.2. Differentials in the Eilenberg-Moore spectral sequence
8.3. Further structure
273
274
297
313
8bis. Nontrivial Fundamental Groups
8b'M. Actions of the fundamental group
8bls.2. Homology of groups
8b's.3. Nilpotent spaces and groups
329
330
334
344
9. The Adams Spectral Sequence
9.1. Motivation: What cohomology sees
9.2. More homological algebra; the functor Ext
9.3. The spectral sequence
9.4. Other geometric applications
9.5. Computations
9.6. Further structure
366
368
376
392
407
415
430
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10. The Bockstein spectral sequence
10.1. The Bockstein spectral sequence
10.2. Other Bockstein spectral sequences
—
455
458
480
Part III: Sins of Omission
11. More Spectral Sequences in Topology
11.1. Spectral sequences for mappings and spaces of mappings
11.2. Spectral sequences and spectra
11.3. Other Adams spectral sequences
11.4. Equivariant matters
11.5. Miscellanea
485
487
487
495
499
501
504
12. Spectral sequences in Algebra, Geometry and Analysis
12.1. Spectral sequences for rings and modules
12.2. Spectral sequences in geometry
12.3. Spectral sequences in algebraic K-theory
12.4. Derived categories
507
507
515
520
523
Bibliography
525
Symbol Index
553
Index
555