Beginner's Course in Topology: Geometric Chapters (Paperback, 1984. 2nd Print)

Set-Theoretical Terms and Notations Used in This Book, but not Generally Adopted.- 1 Topological Spaces.- 1. Fundamental Concepts.- 1. Topologies.- 2. Metrics.- 3. Subspaces.- 4. Continuous Maps.- 5. Separation Axioms.- 6. Countability Axioms.- 7. Compactness.- 2. Constructions.- 1. Sums.- 2. Products.- 3. Quotients.- 4. Glueing.- 5. Projective Spaces.- 6. More Special Constructions.- 7. Spaces of Continuous Maps.- 8. The Case of Pointed Spaces.- 9. Exercises.- 3. Homotopies.- 1. General Definitions.- 2. Paths.- 3. Connectedness and k-Connectedness.- 4. Local Properties.- 5. Borsuk Pairs.- 6. CNRS-Spaces.- 7. Homotopy Properties of Topological Constructions.- 8. Exercises.- 2 Cellular Spaces.- 1. Cellular Spaces and Their Topological Properties.- 1. Fundamental Concepts.- 2. Glueing Cellular Spaces from Balls.- 3. The Canonical Cellular Decompositions of Spheres, Balls, and Projective Spaces.- 4. More Topological Properties of Cellular Spaces.- 5. Cellular Constructions.- 6. Exercises.- 2. Simplicial Spaces.- 1. Euclidean Simplices.- 2. Simplicial Spaces and Simplicial Maps.- 3. Simplicial Schemes.- 4. Polyhedra.- 5. Simplicial Constructions.- 6. Stars. Links. Regular Neighborhoods.- 7. Simplicial Approximation of Continuous Maps.- 8. Exercises.- 3. Homotopy Properties of Cellular Spaces.- 1. Cellular Pairs.- 2. Cellular Approximation of Continuous Maps.- 3. k-Connected Cellular Pairs.- 4. Simplicial Approximation of Cellular Spaces.- 5. Exercises.- 3 Smooth Manifolds.- 1. Fundamental Concepts.- 1. Topological Manifolds.- 2. Differentiable Structures.- 3. Orientations.- 4. The Manifold of Tangent Vectors.- 5. Embeddings, Immersions, and Submersions.- 6. Complex Structures.- 7. Exercises.- 2. Stiefel and Grassman Manifolds.- 1. Stiefel Manifolds.- 2. Grassman Manifolds.- 3. Some Low-Dimensional Stiefel and Grassman Manifolds.- 4. Exercises.- 3. A Digression: Three Theorems from Calculus.- 1. Polynomial Approximation of Functions.- 2. Singular Values.- 3. Nondegenerate Critical Points..- 4. Embeddings. Immersions. Smoothings. Approximations.- 1. Spaces of Smooth Maps.- 2. The Simplest Embedding Theorems.- 3. Transversalizations and Tubes.- 4. Smoothing Maps in the Case of Closed Manifolds.- 5. Glueing Manifolds Smoothly.- 6. Smoothing Maps in the Presence of a Boundary.- 7. General Position.- 8. Maps Transverse to a Submanifold.- 9. Raising the Smoothness Class of a Manifold.- 10. Approximation of Maps by Embeddings and Immersions.- 11. Exercises.- 5. The Simplest Structure Theorems.- 1. Morse Functions.- 2. Cobordisms and Surgery.- 3. Two-dimensional Manifolds.- 4. Exercises.- 4 Bundles.- 1. Bundles without Group Structure.- 1. General Definitions.- 2. Locally Trivial Bundles.- 3. Serre Bundles.- 4. Bundles With Map Spaces as Total Spaces.- 5. Exercises.- 2. A Digression: Topological Groups and Transformation Groups.- 1. Topological Groups.- 2. Groups of Homeomorphisms.- 3. Actions.- 4. Exercises.- 3. Bundles with a Group Structure.- 1. Spaces With F-Structure.- 2. Steenrod Bundles.- 3. Associated Bundles.- 4. Ehresmann-Feldbau Bundles.- 5. Exercises.- 4. The Classification of Steenrod Bundles.- 1. Steenrod Bundles and Homotopies.- 2. Universal Bundles.- 3. The Milnor Bundles.- 4. Reductions of the Structure Group.- 5. Exercises.- 5. Vector Bundles.- 1. General Definitions.- 2. Constructions.- 3. The Classical Universal Vector Bundles.- 4. The Most Important Reductions of the Structure Group.- 5. Exercises.- 6. Smooth Bundles.- 1. Fundamental Concepts.- 2. Smoothings and Approximations.- 3. Smooth Vector Bundles.- 4. Tangent and Normal Bundles.- 5. Degree.- 6. Exercises.- 5 Homotopy Groups.- 1. The General Theory.- 1. Absolute Homotopy Groups.- 2. A Digression: Local Systems.- 3. Local Systems of Homotopy Groups of a Topological Space.- 4. Relative Homotopy Groups.- 5. A Digression: Sequences of Groups and Homomorphisms, and ?-Sequences.- 6. The Homotopy Sequence of a Pair.- 7. The Local System of Homotopy