Preface Notation and Terminology CHAPTER I Two-Dimensional Manifolds ��1��Introduction ��2��Definition and Examples of n-Manifolds ��3��Orientable vs��Nonorientablc Manifolds ��4��Examples of Compact�� Connected 2-Manifolds ��5��Statement of the Classification Theorem for Compact Surfaces ��6��Triangulations ofCompact Surfaces ��7��Proof of Theorem 5��1 ��8��The Euler Characteristic of a Surface References CHAPTER 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 References CHAPTER III Free Groups and Free Products of Groups ��1��Introduction ��2��The Weak Product of Abelian Groups ��3��Free Abelian Groups ��4��Free Products ofGroups ��5��Free Groups ��6��The Presentation of Groups by Generators and Relations ��7��Universal Mapping Problems References CHAPTER 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 References ���� CHAPTER V Covering Spaces CHAPTER VI Background and Motivation for Homology Theory CHAPTER VII Definitions and Basic Properties of Homology Theory CHAPTER VIII Determination of the Homology Groups of Certain Spaces�� Applications and Further Properties of Homology Theory CHAPTER IX Homology of CW-Complexes CHAPTER X Homology with Arbitrary Coefficient Groups CHAPTER XI The Homology of Product Spaces CHAPTER XII Cohomology Theory CHAPTER XIII Products in Homology and Cohomology CHAPTER XIV Duality Theorems for the Homology of Manifolds CHAPTER XV Cup Products in Project we Spaces and Applications of Cup Products APPENDIX Index