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Introduction to Topology

V. A. Vassiliev
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 14
[Reviewed by
Fernando Q. Gouvêa
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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 Rn 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 ([email protected]) 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