Singular Homology Theory: 127 - Rilegato

Massey, William S.

Rilegato

ISBN 10: 038797430X ISBN 13: 9780387974309

Casa editrice: Springer Nature, 1991

Riassunto

This textbook is intended for a course in algebraic topology at the beginning graduate level. The main topics covered are: the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. The text consists of material from the first five chapters of the author's earlier book, "Algebraic Topology; an Introduction (GTM 56)" together with almost all of his book, "Singular Homology Theory (GTM 70)". The material from the two earlier books has been substantially revised, corrected, and brought up to date. This textbook on algebraic topology is intended for graduate students in mathematics.

Su questo libro

L'autore:

William S. Massey (1920-2017) was an American mathematician known for his work in algebraic topology. The Blakers-Massey theorem and the Massey product were both named for him. His textbooks Singular Homology Theory and Algebraic Topology: An Introduction are also in the Graduate Texts in Mathematics series.

Contenuti:

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.

Su questa edizione

• Editore: Springer Nature
• Data di pubblicazione: 1991
• ISBN 10: 038797430X
• ISBN 13: 9780387974309
• Rilegatura: Copertina rigida
• Numero di pagine: 452