## A User's Guide to Spectral SequencesSpectral 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 |

553 | |

555 | |

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### Common terms and phrases

abelian group Adams spectral sequence algebra structure apply bar construction bidegree bigraded algebra Bockstein spectral sequence bundle Chapter coalgebra cochain coefﬁcients coﬁbration compute H consider converging to H Corollary cosimplicial CW-complex deﬁned deﬁnition degree denote determine differential graded algebra differential graded module dimension double complex dual E2-term Eilenberg-Moore spectral sequence elements epimorphism exact couple example ﬁbre ﬁeld ﬁltered ﬁltration ﬁnd ﬁnite type Furthermore geometric given graded commutative graded module graded vector space H-space homological algebra homology homomorphism homotopy groups homotopy theory homotopy type Hopf algebra identiﬁed implies induces an isomorphism isomorphism Lemma Leray-Serre spectral sequence Lie groups long exact sequence manifold Massey products Math morphism multiplication nilpotent nontrivial nonzero projective resolution properties Proposition prove pullback quotient R-module reader result ring satisﬁes short exact sequence simplicial set simply-connected spectral sequence converges Steenrod algebra submodules Suppose topology trivial