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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
William S Massey |

ISBN: | 9781468492316 1468492314 |

OCLC Number: | 853269753 |

Description: | 1 online resource (428 pages). |

Contents: | I Background and Motivation for Homology Theory -- §1. Introduction -- §2. Summary of Some of the Basic Properties of Homology Theory -- §3. Some Examples of Problems Which Motivated the Developement of Homology Theory in the Nineteenth Century -- §4. References to Further Articles on the Background and Motivation for Homology Theory -- Bibliography for Chapter I -- II Definitions and Basic Properties of Homology Theory -- §1. Introduction -- §2. Definition of Cubical Singular Homology Groups -- §3. The Homomorphism Induced by a Continuous Map -- §4. The Homotopy Property of the Induced Homomorphisms -- §5. The Exact Homology Sequence of a Pair -- §6. The Main Properties of Relative Homology Groups -- §7. The Subdivision of Singular Cubes and the Proof of Theorem 6.3 -- III Determination of the Homology Groups of Certain Spaces : Applications and Further Properties of Homology Theory -- §1. Introduction -- §2. Homology Groups of Cells and Spheres Application -- §3. Homology of Finite Graphs -- §4. Homology of Compact Surfaces -- §5. The Mayer -- Vietoris Exact Sequence -- §6. The Jordan -- Brouwer Separation Theorem and Invariance of Domain -- §7. The Relation between the Fundamental Group and the First Homology Group -- Bibliography for Chapter III -- IV Homology of CW-complexes -- §1. Introduction -- §2. Adjoining Cells to a Space -- §3. CW-complexes -- §4. The Homology Groups of a CW-complex -- §5. Incidence Numbers and Orientations of Cells -- §6. Regular CW-complexes -- §7. Determination of Incidence Numbers for a Regular Cell Complex -- §8. Homology Groups of a Pseudomanifold -- Bibliography for Chapter IV -- V Homology with Arbitrary Coefficient Groups -- §1. Introduction -- §2. Chain Complexes -- §3. Definition and Basic Properties of Homology with Arbitrary Coefficients -- §4. Intuitive Geometric Picture of a Cycle with Coefficients in G -- §5. Coefficient Homomorphisms and Coefficient Exact Sequences -- §6. The Universal Coefficient Theorem -- §7. Further Properties of Homology with Arbitrary Coefficients -- Bibliography for Chapter V -- VI The Homology of Product Spaces -- §1. Introduction -- §2. The Product of CW-complexes and the Tensor Product of Chain Complexes §3. The Singular Chain Complex of a Product Space -- §4. The Homology of the Tensor Product of Chain Complexes (The Künneth Theorem) §5. Proof of the Eilenberg -- Zilber Theorem -- §6. Formulas for the Homology Groups of Product Spaces -- Bibliography for Chapter VI -- VII Cohomology Theory -- §1. Introduction -- §2. Definition of Cohomology Groups -- Proofs of the Basic Properties -- §3. Coefficient Homomorphisms and the Bockstein Operator in Cohomology -- §4. The Universal Coefficient Theorem for Cohomology Groups -- §5. Geometric Interpretation of Cochains, Cocycles, etc -- §6. Proof of the Excision Property; the Mayer -- Vietoris Sequence -- Bibliography for Chapter VII -- VIII Products in Homology and Cohomology -- §1. Introduction -- §2. The Inner Product -- §3. An Overall View of the Various Products -- §4. Extension of the Definition of the Various Products to Relative Homology and Cohomology Groups -- §5. Associativity, Commutativity, and Existence of a Unit for the Various Products -- §6. Digression : The Exact Sequence of a Triple or a Triad -- §7. Behavior of Products with Respect to the Boundary and Coboundary Operator of a Pair -- §8. Relations Involving the Inner Product -- §9. Cup and Cap Products in a Product Space -- §10. Remarks on the Coefficients for the Various Products -- The Cohomology Ring -- §11. The Cohomology of Product Spaces (The Künneth Theorem for Cohomology) -- Bibliography for Chapter VIII -- IX Duality Theorems for the Homology of Manifolds -- §1. Introduction -- §2. Orientability and the Existence of Orientations for Manifolds -- §3. Cohomology with Compact Supports -- §4. Statement and Proof of the Poincaré Duality Theorem -- §5. Applications of the Poincaré Duality Theorem to Compact Manifolds -- §6. The Alexander Duality Theorem -- §7. Duality Theorems for Manifolds with Boundary -- §8. Appendix: Proof of Two Lemmas about Cap Products -- Bibliography for Chapter IX -- X Cup Products in Projective Spaces and Applications of Cup Products -- §1. Introduction -- §2. The Projective Spaces -- §3. The Mapping Cylinder and Mapping Cone -- §4. The Hopf Invariant -- Bibliography for Chapter X -- Appendix A Proof of De Rham's Theorem -- §1. Introduction -- §2. Differentiable Singular Chains -- §3. Statement and Proof of De Rham's Theorem -- Bibliography for the Appendix. |

Series Title: | Graduate texts in mathematics, 70. |

Responsibility: | by William S. Massey. |

### Abstract:

The main purpose of this book is to give a systematic treatment of singular homology and cohomology theory. Singular homology and cohomology theory has been the subject of a number of textbooks in the last couple of decades, so the basic outline of the theory is fairly well established.
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