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A basic course in algebraic topology

Author: William S Massey
Publisher: New York : Springer-Verlag, ©1991.
Series: Graduate texts in mathematics, 127.
Edition/Format:   Print book : EnglishView all editions and formats

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  Read more...


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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: William S Massey
ISBN: 038797430X 9780387974309 354097430X 9783540974307
OCLC Number: 22308878
Description: xvi, 428 pages : illustrations ; 25 cm.
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 Kunneth 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 Kunneth 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 Poincare Duality Theorem.- 5. Applications of the Poincare 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, 127.
Responsibility: William S. Massey.
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W.S. Massey A Basic Course in Algebraic Topology "In the minds of many people algebraic topology is a subject which is a ~esoteric, specialized, and disjoint from the overall sweep of mathematical Read more...

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