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Introduction to Topology
V. A. Vassiliev, Independent University of Moscow, Russia
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Student Mathematical Library
2001; 149 pp; softcover
Volume: 14
ISBN-10: 0-8218-2162-8
ISBN-13: 978-0-8218-2162-6
List Price: US$30 Institutional Members: US$24
All Individuals: US\$24
Order Code: STML/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.

Reviews

"The book will be very convenient for those who want to be acquainted with the topic in a short time."

-- European Mathematical Society Newsletter

"A concise treatment of differential and algebraic topology."

-- American Mathematical Monthly

"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."

-- Mathematical Reviews

Table of Contents

• 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
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