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A User's Guide to Spectral Sequences : Cambridge Studies in Advanced Mathematics - John McCleary

A User's Guide to Spectral Sequences

Cambridge Studies in Advanced Mathematics


Published: 23rd October 2001
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Spectral sequences are among the most elegant, most powerful, and most complicated methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first third of the book treats the algebraic foundations for this sort of homological algebra, starting from informal calculations, to give the novice a familiarity with the range of applications possible. The heart of the book is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.

From reviews of the first edition: 'McCleary has undertaken and completed a daunting task; few algebraic topologists would have the courage to even try to write a book such as this. The mathematical community is indebted to him for this achievement!' Bulletin of the AMS '... this guide is a treasure trove ...'. Niew Archief voor Wiskunde

Prefacep. vii
Introductionp. ix
Algebrap. 1
An Informal Introductionp. 3
"There is a spectral sequence ..."p. 3
Lacunary phenomenap. 7
Exploiting further structurep. 9
Working backwardsp. 19
Interpreting the answerp. 23
What is a Spectral Sequence?p. 28
Definitions and basic propertiesp. 28
How does a spectral sequence arise?p. 31
Spectral sequences of algebrasp. 44
Algebraic applicationsp. 46
Convergence of Spectral Sequencesp. 61
On convergencep. 61
Limits and colimitsp. 67
Zeeman's comparison theoremp. 82
Topologyp. 89
Topological Backgroundp. 91
CW-complexesp. 92
Simplicial setsp. 103
Fibrationsp. 109
Hopf algebras and the Steenrod algebrap. 122
The Leray-Serre spectral sequence Ip. 133
Construction of the spectral sequencep. 136
Immediate applicationsp. 140
Appendicesp. 163
The Leray-Serre spectral sequence IIp. 180
A proof of theorem 6.1p. 181
The transgressionp. 185
Classifying spaces and characteristic classesp. 207
Other constructions of the spectral sequencep. 221
The Eilenberg-Moore Spectral Sequence Ip. 232
Differential homological algebrap. 234
Bringing in the topologyp. 248
The Koszul complexp. 257
The homology of quotient spaces of group actionsp. 265
The Eilenberg-Moore Spectral Sequence IIp. 273
On homogeneous spacesp. 274
Differentials in the Eilenberg-Moore spectral sequencep. 297
Further structurep. 313
Nontrivial Fundamental Groupsp. 329
Actions of the fundamental groupp. 330
Homology of groupsp. 334
Nilpotent spaces and groupsp. 344
The Adams Spectral Sequencep. 366
Motivation: What cohomology seesp. 368
More homological algebra; the functor Extp. 376
The spectral sequencep. 392
Other geometric applicationsp. 407
Computationsp. 415
Further structurep. 430
The Bockstein spectral sequencep. 455
The Bockstein spectral sequencep. 458
Other Bockstein spectral sequencesp. 480
Sins of Omissionp. 485
More Spectral Sequences in Topologyp. 487
Spectral sequences for mappings and spaces of mappingsp. 487
Spectral sequences and spectrap. 495
Other Adams spectral sequencesp. 499
Equivariant mattersp. 501
Miscellaneap. 504
Spectral sequences in Algebra, Geometry and Analysisp. 507
Spectral sequences for rings and modulesp. 507
Spectral sequences in geometryp. 515
Spectral sequences in algebraic K-theoryp. 520
Derived categoriesp. 523
Bibliographyp. 525
Symbol Indexp. 553
Indexp. 555
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521567596
ISBN-10: 0521567599
Series: Cambridge Studies in Advanced Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 578
Published: 23rd October 2001
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 3.0
Weight (kg): 0.77
Edition Number: 2
Edition Type: Revised