One of the most marvelous dividends of the revolution in algebraic geometry that started with Alexander Grothendieck in the 1950s is the infusion of, for lack of a better word, categorical methods into all areas that have some intersection with algebraic geometry, including algebraic topology and number theory. Of course it is equally proper to say that given the influence on Grothendieck of Jean Leray, Henri Cartan, Jean-Pierre Serre, and other French topologists, largely Bourbakians, we are really talking about a cross-fertilization: the usual story in Mathematics.
In any case, as a number theorist it has been, and is, a great joy for me to encounter so many things in my field that have intimate connections to algebraic geometry (consider, in fact, the emergence of the modern field of arithmetic geometry) and algebraic topology, in particular all things cohomological. When I was a kid, the natural arithmetical places to encounter cohomology were primarily found in class field theory, what with Artin-Tate and the Herbrand quotient. The Weil conjectures, and Pierre Deligne, did not appear on arithmetical radar screens until much later — at least for me — and today’s developments, in the wake of Andrew Wiles’ slaying of the Fermat, were (what?) the stuff that certain visionaries’ dreams were made of? But today it is an algebraic topological bonanza even for pragmatist users like me, and it is a boon indeed to have good sources available written in the modern idiom.
The book under review, Lectures on Algebraic Topology, by Sergey V. Matveev, has the additional benefit of being expressly geared toward the rookie: “… an introduction to the basic methods of algebraic topology for the beginner. It presents elements of both homology theory and homotopy theory, and includes various applications.”
Well, more is true. The 99 pages of the book split into 60 pages on homology, in which the author includes coverage of elementary degree theory, relative homology, homology axioms (more pointed than Eilenberg-Steenrod), cellular homology, homology with coefficients, cohomology, as well as Lefschetz‘s fixed point theorem and Poincaré duality. The complementary part of Lectures on Algebraic Topology is then devoted to homotopy, and Matveev takes the reader all the way to higher homotopy groups, bundles, and coverings: very solid stuff.
There are problems and exercises throughout the book, and there is a supplement appended giving answers, hints, and solutions.
Lectures on Algebraic Topology doesn’t waste any time in getting to the good stuff, to quote Feynman, and Matveev, while necessarily terse, does the material justice. I think this book is a fine first step on a road that takes the reader to more advanced and even avant garde material as swiftly as possible: it provides a fine foundation for more advanced texts and even provides the reader with a good deal of eminently appropriate titillation.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.