Topology II, Homotopy and Homology, Classical Manifolds

The volume under review consists of three chapters: Introduction to Homotopy Theory (by O. Ya. Viro and D. B. Fuchs), Homology and Cohomology (again by O. Ya. Viro and D. B. Fuchs), and Classical Manifolds (by D. B. Fuchs). The original Russian text was translated by C. J. Shaddock. The presentation of notions and results in all three chapters is really very nice. The first two chapters contain quite detailed exposition, while the third chapter has a more encyclopedia like character. This means that the first two chapters can also serve very well as a textbook. I would even recommend them for this purpose, because the presentation is on one hand a detailed one (as already mentioned), and on the other hand it is not too long. The choice of material is very good, the text is saturated with many examples, and we find here information about further developments and recommendations for further study. Naturally, because this is an encyclopedia, we find no exercises. The last chapter contains information about the topology of classical manifolds, and I do not think that information of this type, in such a compact form and to such an extent, can be found elsewhere. Generally, the whole volume makes a very good impression, and I would say that it is very clearly written.

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