## Handbook of Algebraic TopologyAlgebraic topology (also known as homotopy theory) is a flourishing branch of modern mathematics. It is very much an international subject and this is reflected in the background of the 36 leading experts who have contributed to the Handbook. Written for the reader who already has a grounding in the subject, the volume consists of 27 expository surveys covering the most active areas of research. They provide the researcher with an up-to-date overview of this exciting branch of mathematics. |

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### Contents

73 | |

127 | |

Chapter 4 Introduction to fibrewise homotopy theory | 169 |

Chapter 5 Coherent homotopy over a fixed space | 195 |

Chapter 6 Modem foundations for stable homotopy theory | 213 |

Chapter 7 Completions in algebra and topology | 255 |

Chapter 8 Equivariant stable homotopy theory | 277 |

Chapter 9 The stable homotopy theory of finite complexes | 325 |

Chapter 16 Differential graded algebras in topology | 829 |

Chapter 17 Real and rational homotopy theory | 867 |

Chapter 18 Cohomology of groups | 917 |

Chapter 19 Homotopy theory of Lie groups | 951 |

Chapter 20 Computing v1periodic homotopy groups of spheres and some compact Lie groups | 993 |

Chapter 21 Classifying spaces of compact Lie groups and finite loop spaces | 1049 |

Chapter 22 Hspaces with finiteness conditions | 1095 |

Chapter 23 CoHspaces | 1143 |

Chapter 10 The EHP sequence and periodic homotopy | 397 |

Chapter 11 Introduction to nonconnective ImJtheory | 425 |

Chapter 12 Applications of nonconnective Im Jtheory | 463 |

Chapter 13 Stable homotopy and iterated loop spaces | 505 |

Chapter 14 Stable operations in generalized cohomology | 585 |

Chapter 15 Unstable operations in generalized cohomology | 687 |

Chapter 24 Fibration and product decompositions in nonstable homotopy theory | 1175 |

Chapter 25 Phantom maps | 1209 |

Chapter 26 Walls finiteness obstruction | 1259 |

Chapter 27 LusternikSchnirelmann category | 1293 |

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### Common terms and phrases

abelian group acyclic Adams adjoint Algebraic Topology Amer axioms bundle cell chain complex coaction coefficient cofibration cofibre sequence cohomology theory colimit commutative diagram comodule comonad compact composite connected construction COROLLARY CW-complex defined definition denote diag dimension dual duality E*-algebra element equivariant exact sequence example fibration fibre fibrewise pointed space fibrewise space filtration finite FMod functor G-space G-spectra given gives graded hence homology homomorphism homotopy category homotopy equivalence homotopy groups homotopy theory homotopy type ideal induces an isomorphism invariant inverse isomorphism K-theory Lemma Lie groups loop spaces map f Math model category module monoid morphism multiplication natural transformation nilpotent object obtain operations p-local PROOF proper homotopy properties Proposition quasi-isomorphism quotient R-modules result ring spectrum satisfies Section shows simplicial set SLNM smash product spectra spectral sequence stable homotopy stable homotopy theory structure subgroup suspension theorem trivial weak equivalence

### Popular passages

Page 4 - Y, if there are continuous maps / : X —» Y and g : Y —» X such that the composites fg = ly and gf = lx are the identity maps.

Page 7 - ... spaces is one in which the objects are topological spaces and the 'mappings' are not individual maps but homotopy classes of ordinary maps. The equivalences are the classes with two-sided inverses, and two spaces are of the same homotopy type if and only if they are related by such an equivalence. The ultimate object of algebraic homotopy is to construct a purely algebraic theory, which is equivalent to homotopy theory in the same sort of way that 'analytic' is equivalent to 'pure