Author | Massey, William S. author |
---|---|
Title | Singular Homology Theory [electronic resource] / by William S. Massey |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1980 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9231-6 |
Descript | XVI, 428 p. online resource |
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 MayerโVietoris Exact Sequence -- ยง6. The JordanโBrouwer 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 EilenbergโZilber Theorem -- ยง6. Formulas for the Homology Groups of Product Spaces -- Bibliography for Chapter VI -- VII Cohomology Theory -- ยง1. Introduction -- ยง2. Definition of Cohomology GroupsโProofs 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 MayerโVietoris 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 ProductsโThe 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 Rhamโs Theorem -- ยง1. Introduction -- ยง2. Differentiable Singular Chains -- ยง3. Statement and Proof of De Rhamโs Theorem -- Bibliography for the Appendix