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AuthorMassey, William S. author
TitleSingular Homology Theory [electronic resource] / by William S. Massey
ImprintNew York, NY : Springer New York : Imprint: Springer, 1980
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Descript XVI, 428 p. online resource


The main purpose of this book is to give a systematic treatment of singular homology and cohomology theory. It is in some sense a sequel to the author's previous book in this Springer-Verlag series entitled Algebraic Topology: An Introduction. This earlier book is definitely not a logical prerequisite for the present volume. However, it would certainly be advantageous for a prospective reader to have an acquaintance with some of the topics treated in that earlier volume, such as 2-dimensional manifolds and the fundaยญ mental group. 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. Therefore, from the point of view of the mathematics involved, there can be little that is new or original in a book such as this. On the other hand, there is still room for a great deal of variety and originality in the details of the exposition. In this volume the author has tried to give a straightforward treatment of the subject matter, stripped of all unnecessary definitions, terminology, and technical machinery. He has also tried, wherever feasible, to emphasize the geometric motivation behind the various concepts


I Background and Motivation for Homology Theory -- ยง1. Introduction -- ยง2. Summary of Some of the Basic Properties of Homology Theory -- ยง3. Some Examples of Problems Which Motivated the Developement of Homology Theory in the Nineteenth Century -- ยง4. References to Further Articles on the Background and Motivation for Homology Theory -- Bibliography for Chapter I -- II Definitions and Basic Properties of Homology Theory -- ยง1. Introduction -- ยง2. Definition of Cubical Singular Homology Groups -- ยง3. The Homomorphism Induced by a Continuous Map -- ยง4. The Homotopy Property of the Induced Homomorphisms -- ยง5. The Exact Homology Sequence of a Pair -- ยง6. The Main Properties of Relative Homology Groups -- ยง7. 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 -- ยง1. Introduction -- ยง2. Homology Groups of Cells and Spheres Application -- ยง3. Homology of Finite Graphs -- ยง4. Homology of Compact Surfaces -- ยง5. The Mayerโ{128}{148}Vietoris Exact Sequence -- ยง6. The Jordanโ{128}{148}Brouwer Separation Theorem and Invariance of Domain -- ยง7. The Relation between the Fundamental Group and the First Homology Group -- Bibliography for Chapter III -- IV Homology of CW-complexes -- ยง1. Introduction -- ยง2. Adjoining Cells to a Space -- ยง3. CW-complexes -- ยง4. The Homology Groups of a CW-complex -- ยง5. Incidence Numbers and Orientations of Cells -- ยง6. Regular CW-complexes -- ยง7. Determination of Incidence Numbers for a Regular Cell Complex -- ยง8. Homology Groups of a Pseudomanifold -- Bibliography for Chapter IV -- V Homology with Arbitrary Coefficient Groups -- ยง1. Introduction -- ยง2. Chain Complexes -- ยง3. Definition and Basic Properties of Homology with Arbitrary Coefficients -- ยง4. Intuitive Geometric Picture of a Cycle with Coefficients in G -- ยง5. Coefficient Homomorphisms and Coefficient Exact Sequences -- ยง6. The Universal Coefficient Theorem -- ยง7. Further Properties of Homology with Arbitrary Coefficients -- Bibliography for Chapter V -- VI The Homology of Product Spaces -- ยง1. Introduction -- ยง2. The Product of CW-complexes and the Tensor Product of Chain Complexes ยง3. The Singular Chain Complex of a Product Space -- ยง4. The Homology of the Tensor Product of Chain Complexes (The Kรผnneth Theorem) ยง5. Proof of the Eilenbergโ{128}{148}Zilber Theorem -- ยง6. Formulas for the Homology Groups of Product Spaces -- Bibliography for Chapter VI -- VII Cohomology Theory -- ยง1. Introduction -- ยง2. Definition of Cohomology Groupsโ{128}{148}Proofs of the Basic Properties -- ยง3. Coefficient Homomorphisms and the Bockstein Operator in Cohomology -- ยง4. The Universal Coefficient Theorem for Cohomology Groups -- ยง5. Geometric Interpretation of Cochains, Cocycles, etc -- ยง6. Proof of the Excision Property; the Mayerโ{128}{148}Vietoris Sequence -- Bibliography for Chapter VII -- VIII Products in Homology and Cohomology -- ยง1. Introduction -- ยง2. The Inner Product -- ยง3. An Overall View of the Various Products -- ยง4. Extension of the Definition of the Various Products to Relative Homology and Cohomology Groups -- ยง5. Associativity, Commutativity, and Existence of a Unit for the Various Products -- ยง6. Digression : The Exact Sequence of a Triple or a Triad -- ยง7. Behavior of Products with Respect to the Boundary and Coboundary Operator of a Pair -- ยง8. Relations Involving the Inner Product -- ยง9. Cup and Cap Products in a Product Space -- ยง10. Remarks on the Coefficients for the Various Productsโ{128}{148}The Cohomology Ring -- ยง11. The Cohomology of Product Spaces (The Kรผnneth Theorem for Cohomology) -- Bibliography for Chapter VIII -- IX Duality Theorems for the Homology of Manifolds -- ยง1. Introduction -- ยง2. Orientability and the Existence of Orientations for Manifolds -- ยง3. Cohomology with Compact Supports -- ยง4. Statement and Proof of the Poincarรฉ Duality Theorem -- ยง5. Applications of the Poincarรฉ Duality Theorem to Compact Manifolds -- ยง6. The Alexander Duality Theorem -- ยง7. Duality Theorems for Manifolds with Boundary -- ยง8. Appendix: Proof of Two Lemmas about Cap Products -- Bibliography for Chapter IX -- X Cup Products in Projective Spaces and Applications of Cup Products -- ยง1. Introduction -- ยง2. The Projective Spaces -- ยง3. The Mapping Cylinder and Mapping Cone -- ยง4. The Hopf Invariant -- Bibliography for Chapter X -- Appendix A Proof of De Rhamโ{128}{153}s Theorem -- ยง1. Introduction -- ยง2. Differentiable Singular Chains -- ยง3. Statement and Proof of De Rhamโ{128}{153}s Theorem -- Bibliography for the Appendix

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