This is an exceptional book. It clearly reveals that it is a work of masters. Its main virtues are a wealth of ideas and a great emphasis on important concrete examples. The authors show algebraic topology, in particular homology theory, at work in differential topology, differential geometry, and complex analysis. They wrote this book as a sequel to their book “Modern geometry” [Moskva: “Nauka” (1980; Zbl 0433.53001)]; English translation: Modern geometry - methods and applications [New York: Springer, Vol. I (1984; Zbl 0529.53002), Vol. II (1985; Zbl 0565.57001)] and refer many times to the latter, but a reader with the corresponding background will easily do without it. The ”corresponding background” in this case is basic homotopy theory, basic differential manifold theory with some Riemannian geometry, and basic fiber bundle (or at least vector bundle) theory.
The book is divided into 3 chapters and has two appendices. Chapter 1 introduces homology and cohomology; it gives the definitions and explains the basic properties of and the interrelations among de Rham cohomology, simplicial, cellular, and singular homology and cohomology, homology and cohomology with local coefficients, and sheaf (Čech) cohomology. More advanced calculations and applications include cohomology rings of \(H\)-spaces and Lie groups, homology of bundles, obstructions to extending maps and sections, characteristic classes, cohomology operations, and rational homotopy groups of spheres.
Chapter 2 treats the following topics: Morse theory, Poincaré duality, Lyusternik-Shnirel’man category, calculus of variations on Riemannian manifolds and in particular geodesics, Bott periodicity, and the \(n\)-body problem in the plane. In Chapter 3 the authors discuss smooth cobordism, signature, representability of homology classes by (singular or embedded) smooth manifolds, topological and homotopy invariance of rational Pontryagin classes, \(h\)-cobordism groups of smooth homotopy spheres, and smooth structures on manifolds.
How could the authors pack so much material into 340 pages? Firstly, by giving up exhaustiveness and generality; they avoid homological algebra almost completely and consequently they often restrict themselves to coefficients in a field; various generally valid results are stated and proved only for CW complexes or even only for smooth manifolds. Secondly, by omitting details (especially the more technical ones) of arguments; sometimes they even “cheat” a little by hiding the difficulties from the reader, presumably in order to emphasize the essential simplicity of ideas. Unfortunately, in this principle of working on a large scale they go as far as to become a bit careless: sometimes the reader is forced to guess what they mean, and they give some inconsistent definitions and incorrect statements.
There are about 150 problems for the reader scattered through the text. Their purpose is to supplement the results proved and to provide additional important examples, but they are not an integral part of the text. Most of them are quite challenging, and some are forbiddingly hard (for example, how many readers can be expected to discover the classification of noncompact surfaces - Problem 4 in § 4 - if they are given nothing except the usual classification of closed surfaces ?).
Because of the properties described above, the book cannot be recommended as a first reading on homology, but it is most warmly recommended to more experienced mathematicians.