代數(shù)拓撲基礎教程

內容概要

This book is intended to serve as a textbook 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 (including cup products　and the duality theorems of Poincare and Alexander). It 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). This material from the two earlier books has　been revised, corrected, and brought up to date. There is enough here for a　full-year course.   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. Several applications of the methods　of algebraic topology to concrete geometrical-topological problems are given　(e.g., Brouwer fixed point theorem, Brouwer-Jordan separation theorem,　lnvariance of Domain. Borsuk-Ulam theoremS.

書籍目錄

PrefaceNotation and TerminologyCHAPTER ⅠTwo-Dimensional Manifolds　1. Introduction　2. Definition and Examples of n-Manifolds　3. Orientable vs. Nonorientable Manifolds　4. Examples of Compact, Connected 2-Manifolds　5. Statement of the Classification Theorem for Compact Surfaces　6. Triangulations of Compact Surfaces　7. Proof of Theorem 5.1　8. The Euler Characteristic of a Surface　ReferencesCHAPTER Ⅱ　The Fundamental Group　1. Introduction                          　2. Basic Notation and Terminology　3. Definition of the Fundamental Group of a Space　4. The Effect of a Continuous Mapping on the Fundamental Group　5. The Fundamental Group of a Circle IS Infinite Cyclic　6. Application: The Brouwer Fixed-Point Theorem in Dimension 2　7. The Fundamental Group of a Product Space　8. Homotopy Type and Homotopy Equivalence of Spaces  ReferencesCHAPTER Ⅲ Free Groups and Free Products of Groups　1. Introduction　2. The Weak Product of Abelian Groups　3. Free Abelian Groups　4. Free Products of Groups　5. Free Groups　6. The Presentation of Groups by Generators and Relations　7. Universal Mapping Problems　ReferencesCHAPTER Ⅳ  Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces. Applications　1. Introduction　2. Statement and Proof of the Theorem of Seifert and Van Kampen　3. First Application of Theorem 2.1　4. Second Application of Theorem 2.1　5. Structure of the Fundamental Group of a Compact Surface　6. Application to Knot Theory　7. Proof of Lemma 2.4　ReferencesCHAPTER Ⅴ Covering Spaces　1. Introduction　2. Definition and Some Examples of Covering Spaces　3. Lifting of Paths to a Covering Space　4. The Fundamental Group of a Covering Space　5. Lifting of Arbitrary Maps to a Covering Space　6. Homomorphisms and Automorphisms of Covering Spaces……

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