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Publisher:

American Mathematical Society

Publication Date:

2001

Number of Pages:

149

Format:

Paperback

Series:

Student Mathematical Library 14

Price:

26.00

ISBN:

978-0-8218-2162-6

Category:

Textbook

[Reviewed by , on ]

Fernando Q. Gouvêa

08/15/2001

V. A. Vassiliev explains that his *Introduction to Topology* contains the lecture notes for a course he has taught several times at the Independent University of Moscow. It must have been quite a course. 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 duality, and more. The book emphasizes intuitive arguments whenever possible, but it also assumes a lot. For example, it assumes that the reader has a good algebra background and is willing to fill in a lot of the detail.

To make this more concrete, here is what is on pages 1 and 2: a short introduction, the definition of a topology, the discrete topology and the topology on a metric space as examples of topologies, the definition of a basis of a topology, an example of an "exotic" topology on **R**, the definition of closed sets and of the closure of a set, and an exercise: "Consider the basis consisting of parallelepipeds in **R**^{n} with edges parallel to the coordinate axes. Is the topology it determines a different one?"

Though this is part of the AMS Student Mathematical Library, it's hard to imagine an undergraduate who can *learn* topology from this book. On the other hand, I can imagine an advanced undergraduate enjoying the book as 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.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of **FOCUS** and **MAA Online**. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

- 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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