A User's Guide to Spectral Sequences

by John McCleary

• ISBN13: 9780521561419
• ISBN10: 0521561418
• Edition: 2nd
• Format: Hardcover
• Copyright: 2000-12-04
• Publisher: Cambridge University Press

Summary

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.

Author Biography

John McCleary is Professor of Mathematics at Vassar College.

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	(240)
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 classes	207	(14)
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 actions	265	(8)
The Eilenberg-Moore Spectral Sequence II	273	(56)
On homogeneous spaces	274	(23)
Differentials in the Eilenberg-Moore spectral sequence	297	(16)
Further structure	313	(16)
Nontrivial Fundamental Groups	329	(156)
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 of mappings	487	(8)
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 Analysis	507	(18)
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