Customer Reviews

Lectures on Algebraic Topology (EMS Series of Lectures in Mathematics) (English and Russian Edition)

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S. Matveev wrote an excellent book for those who wish to quickly familiarize themselves with the basics of algebraic topology and be able to use it in other fields, such as analysis and differential geometry. Everything about this book is fresh and so different from other basic algebraic topology books. Firstly, it is reader-friendly and it gives plenty of geometric motivation. Secondly, unlike its 300 or 400 page counterparts, it does not engage in exhaustive treatment of topics. Thirdly, it contains plethora of lovely pictures that directly appeal to reader's geometric intuition. Finally, a superb translation by E. Pervova, smoothly brings the best features of author's style to the reader.

In only about 80 pages, Matveev leads the reader from the definition of the standard n-simplex to the classification of covering spaces. The starting point is a very careful and yet efficient treatment of simplicial homology. Here, on of the high points of Matveev's presentation is a clear expalantion of algorithms for computing simplicial homology groups. Additionally, the exact sequence of a pair, Meyer-Vietoris sequence and excision are all thoroughly explained and illustrated by well chosen examples.

Next, the author moves to cellular homology, and again does a marvelous job in presenting the essence of computations. Matveev assumes very little abstract algebra. To spare the reader from searching through exhaustive treatments of necessary algebra done elsewhere, he provides lovely algebra refreshers on the following topics: finitely generated abelian groups, tensor product and torsion product, and group presentation. Having allowed the reader to get the feel for fomology, Matveev turns to axioms of homology and indicates what singular homology is. Lastly, he gives careful attention to the universal coefficient theorem for homology.

In the homology part of the book, as an illustration of the covered material, Matveev skillfully injects related topics, such as the degree of a map and Lefschetz fixed point theorem.

Having explained homology, Matveev turns to cohomology, whose treatment culminates in the presentation of Kunneth formula and products in cohomology. The cohomology part ends with Poincare duality.

In the last part of the book, Matveev turns to the basics of homotopy theory and related topics: the fundamental group and van Kampen's theorem, higher homotopy groups and the corresponding exact sequence of higher homotopy groups for a locally trivial bundle, Hurewicz homomorphism, and classification of covering spaces. Again the reader will greatly profit from Matveev's masterful appeal to geometric intuition.

The book contains over 100 exercises, all supplied with hints, complete solutions or answers. This will make the book a great resource for self-study. Most importantly, the reader who embarks on the road of self-study guided by this book, will be able to achieve solid competence in the basics of algebraic topology in a relatively short time. Furthermore, the reader who has a good working knowledge of Matveev's book should be able to read Milnor's Characteristic Classes and Husemoller's Fibre Budles.

There are a lot of basic algebraic topology books on the market, such as Hatcher, Munkres, Vick, Rotman, Bredon, Massey, Greenberg, and Maunder. Most (if not all) of these books give lengthy treatments of topics from Matveev's books (not necessarily more topics but perhaps more in-depth treatment). Naturally, a reader who wishes to specialize in algebraic topology will need to completely digest at least one of the afore-mentioned books. In any event, it seems that having Matveev's book on the side will help. Perhaps getting an all-around picture of the subject from Matveev first and then, if necessary, plunging into one of those lengthier treatises would be a resonable way to approach algebraic topology.

In conclusion, I would like to invite readers to give serious consideration to this book. You will be captivated by Matveev's simplicity, efficiency, and clarity. Perhaps most importantly, you will gain a better appreciation of algebraic topology and acquire much needed skills to reach higher ground in the subject.