Introduction to Topology (Student Mathematical Library, V. 14)

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." Readership: Graduate students, research mathematicians, and theoretical physicists. In little over 140 pages, the book goes all the way from the definition of a topological space to homology and cohomology theory, Morse theory, Poincaré theory, and more ... emphasizes intuitive arguments whenever possible ... a broad survey of the field. It is often useful to have an overall picture of a subject before engaging it in detail. For that, this book would be a good choice." -- MAA Online From a review of the Russian edition ... "The book is based on a course given by the author in 1996 to first and second year students at Independent Moscow University ... the emphasis is on illustrating what is happening in topology, and the proofs (or their ideas) covered are those which either have important generalizations or are useful in explaining important concepts ... This is an excellent book and one can gain a great deal by reading it. The material, normally requiring several volumes, is covered in 123 pages, allowing the reader to appreciate the interaction between basic concepts of algebraic and differential topology without being buried in minutiae." "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.''--Résumé de l'éditeur Vassiliev, here presented in English translation, first presented this material in a 1996 lecture course at Independent Moscow U. The central topic of the classical notions and methods of differential and algebraic topology are distilled into a pithy volume for researchers, specialists, and advanced students in mathematics and theoretical physics. The volume includes treatment of manifolds, cell spaces, coverings and fibrations, homotopy groups, homology and cohomology, intersection index, Morse theory, and PoincarT duality. Annotation c. Book News, Inc., Portland, OR (booknews.com) Topological spaces and operations with them Homotopy groups and homotopy equivalence Coverings Cell spaces (CW-complexes) Relative homotopy groups and the exact sequence of a pair Fiber bundles Smooth manifolds The degree of a map Homology: Basic definitions and examples Main properties of singular homology groups and their computation Homology of cell spaces Morse theory Cohomology and Poincaré duality Some applications of homology theory Multiplication in cohomology (and homology) Index of notations Subject index