A USER'S GUIDE TO SPECTRAL SEQUENCES CAMBRIDGE
Contents
1. A USER S GUIDE TO SPECTRAL SEQUENCES Second Edition JOHN McCLEARY Vassar College CAMBRIDGE G UNIVERSITY PRESS Table of Contents Preface Introduction 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 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 3 Convergence of Spectral Sequences 3 1 On convergence 3 2 Limits and colimits 3 3 Zeeman s comparison theorem 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 5 The Leray Serre spectral sequence I 5 1 Construction of the spectral sequence 5 2 Immediate applications 5 3 Appendices 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 7 The Eilenberg Moore Spectral Sequence I 7 1 Differential homological algebra 7 2 Bringing in the topology 103 109 122 133 136 140 163 180 181 185 207 221 232 234 248 Table of Contents 7 3 The Koszul complex 7 4 The homology of quotient spaces of group actions 8 The E2. ilenberg Moore Spectral Sequence H 8 1 On homogeneous spaces 8 2 Differentials in the Eilenberg Moore spectral sequence 8 3 Further structure gts Nontrivial Fundamental Groups seis 1 Actions of the fundamental group geis 2 Homology of groups ges 3 Nilpotent spaces and groups 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 10 The Bockstein spectral sequence _ 10 1 The Bockstein spectral sequence 10 2 Other Bockstein spectral sequences 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 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 Bibliography Symbol Index Index XV 257 265 273 274 297 313 329 330 334 344 366 368 376 392 407 415 430 455 458 480 485 487 487 495 499 501 504 507 507 515 520 523 525 553 555
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