Algebraic topology: A first course

Description

The main goal of this book is to serve as an introduction to algebraic topology for undergraduates. It is a sad fact that, because a thorough study of algebraic topology requires a working knowledge of other mathematical disciplines, the average mathematics student usually does not encounter this beautiful subject until he reaches graduate school. This postponement is no longer justified since undergraduates nowadays learn much more than they did in the past. Therefore, the author believes that this book will be quite accessible to seniors or advanced juniors on the undergraduate level. In addition, it should also be of interest to many beginning graduate students. Most graduate courses on algebraic topology these days develop homology theory in a rather abstract and axiom- atic manner. This is certainly the efficient and elegant way to present the material and it also brings the student to current research in the quickest way possible. Furthermore, it is a natural phenomenon in mathematics and other disciplines that with the passage of time the presentation of a theory is tidied up and perhaps significant improvements are made in it. Nevertheless, the beautiful intuition which gave rise to the theory in the first place should not be forgotten. On a number of occasions the author has witnessed how the abstract approach in first-year graduate courses, by hiding the geometry involved, has left students with the feeling that they have merely learned to manipulate some definitions. Thus, the fact that this book tr1es to place geometrical considerations first, and emphasize the historical development, should make it a valuable reference for graduate students also. This book grew out of some notes for a semester course on geometry taught by the author on two separate occasions to third- year college students--once at Wesleyan University and once at the University of Auckland in New Zealand. In this one semester it was possible to cover most of Chapters 1 to 5 and part of Chapter 6. Students with only a background in calculus and linear algebra were able to manage the Material quite well. Actually, nothing but the most basic concepts associated to vector spaces, such as linear independence, basis, and dimension, are used in a few places. On the other hand, a background in linear algebra is helpful in understanding the group aspect of homology theory. The elementary facts of the theory of abelian groups which are used in this book are summarized in Appendix B. The only hard theorem is the Fundamental Theorem of Finitely Generated Abelian Groups, and that can be assumed. A brief discussion of the material in Appendix B by the instructor should be quite sufficient for the students. Appendix A summarizes the facts needed about Jif and continuous maps. Here and there, but mainly in Section 7.4, a little acquaintance with complex numbers comes in handy. It is recommended that the reader glance over both Appendix A and B before starting on the book because they introduce some basic notation which is used throughout the text. Although the concept of an abstract topological space is an important one and indispensable to modern topology, we shall restrict ourselves entirely to subspaces of Jif in this book, thereby avoiding a lengthy discussion of general topology that would only slightly simplify some of the arguments while at the same time undermining the overall goal, which is to display some of the beautiful geometric insights of topology without burdening the uninitiated student with an overpowering load of technical details. One minor exception to this goal of keeping our presentation geometric is the material on abstract complexes in Section 2.3. This can be skipped because one could easily leave cutting and pasting techniques on the intuitive level. Another exception is Theorem 1 in Section 5.4. This nice result is used to define the important induced homomorphis on homology, but the proof involves some rather abstract ideas which were difficult to avoid. Depending on the level of the class, the instructor may decide simply to sketch the proof or skip it alto- gether. It is included mainly for the sake of completeness. The reader is urged to consult the excellent surveys [De-H], [Stnz], and (Ti-V] for additional historical information. The autho in no way claims any completeness in this regard. The texts (Se-T 1 and [Alexf-H] are also highly recommended reading. There is no author index but one can use the bibliography instead because it giv the page numbers where the author's name appears. The birth dates of most authors in the bibliography are usually given where they first appear in the book. Front Cover Contents Preface 1: SOME OLD TOPOLOGICAL PROBLEMS 1.0. Introduction 1.1. The Descartes-Euler Theorem 1.2. Coloring Maps and Graphs 1.3. The Jordan Curve Theorem; Knots 1.4. Further Early Investigations 2: THE GEOMETRY OF COMPLEXES 2.1. What is Topology 2.2. Simplicial Complexes and Maps; Polyhedra 2.3. Abstract Simplicial Complexes; Cutting and Pasting 2.4. Historical Comments 3: THE CLASSIFICATION OF SURFACES 3.0. Introduction 3.1. The Definition of a Surface 3.2. Representing Surfaces by Symbols 3.3. The Normal Form for Some Surfaces 3.4. The Classification Theorems 3.5. Bordered and Noncompact Surfaces 3.6. Historical Comments 4: THE HOMOLOGY GROUPS 4.1. Some Motivation 4.2. The Orientation of a Simplex 4.3. The Definition of the Homology Groups 4.4. The Homology Groups of a Cone 4.5. The Topological Invariance of Homology Groups 4.6. Historical Comments 5: MAPS AND HOMOTOPY 5.1. Simplicial Maps Again 5.2. Homotopy 5.3. Simplicial Approximations 5.4. The Barycentric Subdivision 5.5. The Simplicial Approximation Theorem; Induced Homomorphisms 5.6. Historical Comments 6: FIRST APPLICATIONS OF HOMOLOGY THEORY 6.1. A Quick Review 6.2. Local Homology Groups 6.3. Some Invariance Theorems 6.4. Homology with Arbitrary Coefficients; The Mod 2 Homology Groups 6.5. Pseudomanifolds; Orientability 6.6. Euler's Theorem Revisited 6.7. Retracts and the Brouwer Fixed-Point Theorem 6.8. Historical Comment 7: MAPS OF SPHERES AND MORE APPLICATIONS 7.1. The Degree of a Map; Vector Fields 7.2. The Borsuk-Ulam Theorem and the Ham Sandwich Problem 7.3. More on Degrees of Maps; Zeros of Polynomials More on Degrees of Maps; Zeros of Polynomials 7.4. Local Degrees, Solvability of Equations and Some Complex Analysis 7.5. Extending Maps 7.6. The Jordan Curve Theorem and Other Separation Theorems 7.7. Historical Comments 8: CONCLUDING REMARKS APPENDIX A: The Topology of R^n APPENDIX B: Permutations and Abelian Groups APPENDIX C: The Incidence Matrices List of Symbols Bibliography Index