This is a comprehensive three-volumes-in-one introduction to combinatorial topology by one of the masters. Dover chose to publish the three volumes, which originally appeared in English translation in 1957–1960, bound as one with separate pagination and tables of content. It is probably fair to say that the author, building on work of Hausdorff and Fréchet, contributed in major ways to the development of topology. His colleagues in this endeavor included Urysohn, Heinz Hopf and Brouwer. Alexandrov is considered by many to be responsible for establishing the foundations of homology theory.
The first volume of the three begins with a summary of point set topology and works its way to the basics of simplicial topology and dimension theory. Along the way there is a proof of the Jordan Curve Theorem and classification of closed surfaces. The second volume focuses on a complete development of the theory of Betti groups. In the third volume, the shortest of the three, Alexandrov treats homology manifolds (connected n-dimensional polyhedra whose local Betti groups are isomorphic to local Betti groups of Euclidean space of the same dimension) as well as cohomology groups, duality and fixed point theorems.
Alexandrov is fairly blunt in his assessment of the book in the preface: “… a reader can become acquainted with the ideas of modern topology only by a detailed study of the fundamental topological facts. I have endeavored to present these facts together with the necessary technical apparatus, often cumbersome and not at all attractive, with all logical rigor and at the risk of being boring and tiresome at times.” This is a fair assessment, as far as it goes. What this book has that the slicker axiomatic treatments do not is a concrete, hands-on feeling for homology theory.
The book was written as a text for graduate students specializing in topology or in a related field. It would not today be a textbook of choice, but it gives a unique view of the development of the subject.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.