Beginner's Course in Topology : Geometric Chapters

This book is the result of reworking part of a rather lengthy course of lectures of which we delivered several versions at the Leningrad and Moscow Universities. In these lectures we presented an introduction to the fundamental topics of topology: homology theory, homotopy theory, theory of bundles, and topology of manifolds. The structure of the course was well determined by the guiding term elementary topology, whose main significance resides in the fact that it made us use a rather simple apparatus. tn this book we have retained {hose sections of the course where algebra plays a subordinate role. We plan to publish the more algebraic part of the lectures as a separate book. Reprocessing the lectures to produce the book resulted in the profits and losses inherent in such a situation: the rigour has increased to the detriment of the intuitiveness, the geometric descriptions have been replaced by formulas needing interpretations, etc. Nevertheless, it seems to us tha·t the book retains the main qualities of our lectures: their elementary, systematic, and pedagogical features. The preparation of the reader is assumed to be limi ted to the usual knowledge of set ·theory, algebra, and calculus which mathematics students should master after the first year and a half of studies. The exposition is accompanied by examples and exercises. We hope that the book can be used as a topology textbook.

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 Groups of the Fibers of a Serre Bundle.- 8. The Homotopy Sequence of a Serre Bundle.- 9. The Influence of Other Structures Upon Homotopy Groups.- 10. Alternative Descriptions of the Homotopy Groups.- 11. Additional Theorems.- 12. Exercises.- §2. The Homotopy Groups of Spheres and of Classical Manifolds.- 1. Suspension in the Homotopy Groups of Spheres.- 2. The Simplest Homotopy Groups of Spheres.- 3. The Composition Product.- 4. Information: Homotopy Groups of Spheres.- 5. The Homotopy Groups of Projective Spaces and Lenses.- 6. The Homotopy Groups of Classical Groups.- 7. The Homotopy Groups of Stiefel Manifolds and Spaces.- 8. The Homotopy Groups of Grassman Manifolds and Spaces.- 9. Exercises.- §3. Homotopy Groups of Cellular Spaces.- 1. The Homotopy Groups of One-dimensional Cellular Spaces.- 2. The Effect of Attaching Balls.- 3. The Fundamental Group of a Cellular Space.- 4. Homotopy Groups of Compact Surfaces.- 5. The Homotopy Groups of Bouquets.- 6. The Homotopy Groups of a k-Connected Cellular Pair.- 7. Spaces With Prescribed Homotopy Groups.- 8. Eight Instructive Examples.- 9. Exercises.- §4. Weak Homotopy Equivalence.- 1. Fundamental Concepts.- 2. Weak Homotopy Equivalence and Constructions.- 3. Cellular Approximations of Topological Spaces.- 4. Exercises.- §5. The Whitehead Product.- 1. The Class wd(m,n).- 2. Definition and the Simplest Properties of the Whitehead Product.- 3. Applications.- 4. Exercises.- §6. Continuation of the Theory of Bundles.- 1. Weak Homotopy Equivalence and Steenrod Bundles.- 2. Theory of Coverings.- 3. Orientations.- 4. Some Bundles Over Spheres.- 5. Exercises.- Glossary of Symbols.