Singular Homology Theory

ISBN-13: 9781468492330 / Angielski / Miękka / 2012 / 428 str.

W. S. Massey

Opis

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 funda- 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.

Spis treści

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.

Blurb

This textbook on homology and cohomology theory is geared towards the beginning graduate student. Singular homology theory is developed systematically, avoiding all unnecessary definitions, terminology, and technical machinery. Wherever possible, the geometric motivation behind various algebraic concepts is emphasized.

The only formal prerequisites are knowledge of the basic facts of abelian groups and point set topology. Singular Homology Theory is a continuation of t he author's earlier book, Algebraic Topology: An Introduction, which presents such important supplementary material as the theory of the fundamental group and a thorough discussion of 2-dimensional manifolds. However, this earlier book is not a prerequisite for understanding Singular Homology Theory.