This is a textbook on algebraic topology. It covers all topics now considered to be standard knowledge in the field: basic homotopy theory, (co)homology, duality theorems and also characteristic classes and bordism. The author's viewpoint is that homotopy is more basic than homology. He starts by explaining fundamental groups and covering spaces. After the necessary preliminaries (suspension, (co)fibration, etc.), basic properties of homotopy groups are derived, followed by a brief exposition of stable homotopy. Homology is first introduced abstractly via spectra for pointed spaces and then by an explicit construction of singular homology. The theory is applied for cell complexes and manifolds to get results on cellular homology, fundamental class, winding numbers, etc. An exposition of cohomology (the cup product, Thom isomorphism and the Leray-Hirsch theorem) precedes a chapter on Poincaré duality. The chapter on characteristic classes emphasises the role of classifying spaces. An interplay between homotopy and homology (including the Hurewicz theorem, cohomology of Eilenberg-Mac Lane spaces and homotopy groups of spheres) is illustrated briefly. The last chapter is devoted to bordism as a homology theory. The theory is carefully built and the book may serve as an extensive source for references. The mainstream of the exposition is occasionally supplemented by examples going beyond topology. The reader can also find many exercises here. There are only minimal prerequisites: elementary point set topology, algebra and category theory. Many concepts, such as cell complexes, manifolds and bundles, are explained from scratch.

Reviewer:

md