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Frontmatter -- CONTENTS -- INTRODUCTION -- § 1. Analytic and Algebraic Topology -- § 2. Problems and Examples -- PART I. SIMPLICIAL COMPLEXES -- Chapter 1. GEOMETRY OF SIMPLICIAL COMPLEXES -- § 3. Hulls and Stars -- § 4. Barycentric Stars -- § 5. Simplicial Mappings -- § 6. Neighboring Mappings -- Chapter 2. HOMOLOGY GROUPS AND COHOMOLOGY GROUPS -- § 7. Orientation. Incidence Numbers -- § 8. Homology Groups -- § 9. Examples and Applications -- § 10. Cohomology Groups -- § 11. Homotopic Mappings -- PART II. CHAIN COMPLEXES AND THEIR APPLICATIONS -- Chapter 3. GENERAL THEORY -- § 12. Homology Groups of Chain Complexes -- § 13. Subcomplexes and Factor Complexes -- § 14. The Boundary Operator -- Chapter 4. FREE CHAIN COMPLEXES -- § 15. Modules and Dual Modules -- § 16. Mappings and Dual Mappings -- § 17. Free Chain Complexes. Canonical Bases -- PART III. CELL COMPLEXES. INVARIANCE -- Chapter 5. CELL COMPLEXES -- § 18. Cell Decompositions -- § 19. The Homology Groups of Cell Decompositions -- § 20. Normal Subdivisions -- Chapter 6. INVARIANCE OF THE HOMOLOGY GROUPS -- § 21. Proof of Invariance -- § 22. Supplements. Generalizations -- § 23. Results and Applications -- § 24. Local Homology Groups -- PART IV. DEVELOPMENT OF THE THEORY -- Chapter 7. PRODUCTS IN POLYHEDRA -- § 25. The Cohomology Ring -- § 26. The Cap Product -- Chapter 8. MANIFOLDS -- § 27. Definitions -- § 28. Complementary Cell Decompositions -- § 29. The Poincaré Duality Theorem -- Chapter 9. THE COHOMOLOGY RING OF A MANIFOLD -- § 30. Products in Manifolds -- § 31. Product Matrices -- BIBLIOGRAPHY -- INDEX