スペクトル系列のユーザーズガイド 第2版<br>A User's Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics) (2 SUB)

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スペクトル系列のユーザーズガイド 第2版
A User's Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics) (2 SUB)

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  • 製本 Hardcover:ハードカバー版/ページ数 561 p.
  • 言語 ENG
  • 商品コード 9780521561419
  • DDC分類 514.2

基本説明

Introduces the algebraic foundations in an accessible manner, starting from informal calculations, to give the novice a familiarity with the range of applications possible with spectral sequences.

Full Description


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.

Table of Contents

Preface                                            vii
Introduction ix
Part I: Algebra 1 (88)
An Informal Introduction 3 (25)
``There is a spectral sequence...'' 3 (4)
Lacunary phenomena 7 (2)
Exploiting further structure 9 (10)
Working backwards 19 (4)
Interpreting the answer 23 (5)
What is a Spectral Sequence? 28 (33)
Definitions and basic properties 28 (3)
How does a spectral sequence arise? 31 (13)
Spectral sequences of algebras 44 (2)
Algebraic applications 46 (15)
Convergence of Spectral Sequences 61 (28)
On convergence 61 (6)
Limits and colimits 67 (15)
Zeeman's comparison theorem 82 (7)
Part II: Topology 89 (396)
Topological Background 91 (42)
CW-complexes 92 (11)
Simplicial sets 103(6)
Fibrations 109(13)
Hopf algebras and the Steenrod algebra 122(11)
The Leray-Serre spectral sequence I 133(47)
Construction of the spectral sequence 136(4)
Immediate applications 140(23)
Appendices 163(17)
The Leray-Serre Spectral sequence II 180(52)
A proof of theorem 6.1 181(4)
The transgression 185(22)
Classifying spaces and characteristic 207(14)
classes
Other constructions of the spectral sequence 221(11)
The Eilenberg-Moore Spectral Sequence I 232(41)
Differential homological algebra 234(14)
Bringing in the topology 248(9)
The Koszul complex 257(8)
The homology of quotient spaces of group 265(8)
actions
The Eilenberg-Moore Spectral Sequence II 273(56)
On homogeneous spaces 274(23)
Differentials in the Eilenberg-Moore 297(16)
spectral sequence
Further structure 313(16)
Nontrivial Fundamental Groups 329(37)
Actions of the fundamental group 330(4)
Homology of groups 334(10)
Nilpotent spaces and groups 344(22)
The Adams Spectral Sequence 366(89)
Motivation: What cohomology sees 368(8)
More homological algebra; the functor Ext 376(16)
The spectral sequence 392(15)
Other geometric applications 407(8)
Computations 415(15)
Further structure 430(25)
The Bockstein spectral sequence 455(30)
The Bockstein spectral sequence 458(22)
Other Bockstein spectral sequences 480(5)
Part III: Sins of Omission 485(40)
More Spectral Sequences in Topology 487(20)
Spectral sequences for mappings and spaces 487(8)
of mappings
Spectral sequences and spectra 495(4)
Other Adams spectral sequences 499(2)
Equivariant matters 501(3)
Miscellanea 504(3)
Spectral sequences in Algebra, Geometry and 507(18)
Analysis
Spectral sequences for rings and modules 507(8)
Spectral sequences in geometry 515(5)
Spectral sequences in algebraic K-theory 520(3)
Derived categories 523(2)
Bibliography 525(28)
Symbol Index 553(2)
Index 555