Description
This English translation of a Russian book presents the basic notions of differential and algebraic topology, which are indispensable for specialists and useful for research mathematicians and theoretical physicists. In particular, ideas and results are introduced related to manifolds, cell spaces, coverings and fibrations, homotopy groups, intersection index, etc. The author notes, “The lecture note origins of the book left a significant imprint on its style. It contains very few detailed proofs: I tried to give as many illustrations as possible and to show what really occurs in topology, not always explaining why it occurs.” He concludes, “As a rule, only those proofs (or sketches of proofs) that are interesting per se and have important generalizations are presented.”
Chapter
Chapter 1. Topological spaces and operations with them
§1.1.Topological spaces and Homeomorphisms
§1.2.Topological operations on topological spaces
Chapter 2. Homotopy groups and homotopy equivalence
§2.1.The fundamental group of a topology space
§2.2.Higher homotopy groups
Chapter 4. Cell spaces (CW-complexes)
§4.1.Definition and main properties of cell spaces
§4.2.Classification of coverings
Chapter 5. Relative homotopy groups and the exact sequence of a pair
§6.1. Locally trivial bundles
§6.2. The exact sequence of a fiber bundle
Chapter 7. Smooth manifolds
§7.3. Tangent bundles over smooth manifolds
§7.4. Riemannian structures
Chapter 8. The degree of a map
§8.1. Critical sets of smooth maps
§8.2. The degree of a map
§8.3. The classification of maps M[sup(n)] -> S[sup(n)]
§8.4. The index of a vector field
Chapter 9. Homology: Basic definitions and examples
§9.1. Chain complexes and their homology
§9.2. Simplicial homology of simplicial polyhedra
Chapter 10. Main properties of singular homology groups and their computation
§10.1. Homology of the point
§10.2. The exact sequence of a pair
§10.3. The exact sequence of a triple
§10.4. Homology of suspensions
§10.5. The Mayer–Vietoris sequence
§10.6. Homolopy of wedges
§10.7. Functoriality of homology
Chapter 11. Homology of cell spaces
§11.1. Cellular complexes
§11.2. Example: homology of projective spaces
§11.3. Cell decomposition of Grassmann manifolds
§12.2. The cellular structure of a manifold endowed with a Morse function
§12.4. Regular Morse functions
§12.5. The boundary operator in a Morse complex
§12.6. Morse inequalities
§12.7. Standard bifurcations of Morse functions
Chapter 13. Cohomology and Poincaré duality
§13.2. Poincaré duality for manifolds without boundary
§13.3. Manifolds with boundary and noncompact manifolds
§13.4. Nonorientable manifolds
Chapter 14. Some applications of homology theory
§14.1. The Hopf invariant
§14.2. The degree of a map
§14.3. The total index of a vector field equals the Euler characteristic
Chapter 15. Multiplication in cohomology (and homology)
§15.1. Homology and cohomology groups of a Cartesian product
§15.2. Multiplication in cohomology
§15.3. Examples of multiplication in cohomology and its geometric meaning
§15.4. Main properties of multiplication in cohomology
§15.5. Connection with the de Rham cohomology
§15.6. Pontryagin multiplication
Introduction to Topology
( Student Mathematical Library )
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