A User's Guide to Spectral Sequences

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Cambridge University Press, 2001 - Mathematics - 561 pages
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Spectral sequences are among the most elegant and powerful 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 part treats the algebraic foundations for this sort of homological algebra, starting from informal calculations. The heart of the text 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.
 

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Contents

1 An Informal Introduction
3
2 What is a Spectral Sequence?
28
3 Convergence of Spectral Sequences
61
Topology
89
4 Topological Background
91
5 The LeraySerre spectral sequence I
133
6 The LeraySerre spectral sequence II
180
7 The EilenbergMoore Spectral Sequence I
232
9 The Adams Spectral Sequence
366
10 The Bockstein spectral sequence
455
Sins of Omission
485
11 More Spectral Sequences in Topology
487
12 Spectral sequences in Algebra Geometry and Analysis
507
Bibliography
525
Symbol Index
553
Index
555

8 The EilenbergMoore Spectral Sequence II
273
8bis Nontrivial Fundamental Groups
329

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About the author (2001)

John McCleary is Professor of Mathematics at Vassar College on the Elizabeth Stillman Williams Chair. His research interests lie at the boundary between geometry and topology, especially where algebraic topology plays a role. His papers on topology have appeared in Inventiones Mathematicae, the American Journal of Mathematics and other journals, and he has written expository papers that have appeared in American Mathematical Monthly. He is also interested in the history of mathematics, especially the history of geometry in the nineteenth century and of topology in the twentieth century. He is the author of Geometry from a Differentiable Viewpoint and A First Course in Topology: Continuity and Dimension and he has edited proceedings in topology and in history, as well as a volume of the collected works of John Milnor. He has been a visitor to the mathematics institutes in Goettingen, Strasbourg and Cambridge, and to MSRI in Berkeley.

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