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Fundamentals of algebraic topology. (English) Zbl 1305.55001
Graduate Texts in Mathematics 270. Cham: Springer (ISBN 978-1-4939-1843-0/hbk; 978-1-4939-1844-7/ebook). x, 163 p. (2014).
This new booklet by the renowned textbook author Steven H. Weintraub is to serve as a quick guide to the fundamental concepts and results of classical algebraic topology. With this explicitly declared objective in mind, he has not striven for presenting just another systematic, comprehensive and detailed introductory textbook on the subject, of which there is already a huge abundance in both the classical and the contemporary literature. On the contrary, he has attempted to give an utmost rapid and concise exposition of the standard topics in algebraic topology, thereby emphasizing the meanwhile classical axiomatic approach developed by S. Eilenberg and N. Steenrod [Foundations of algebraic topology. (Princeton Mathematical Series No.15). Princeton: University Press, XIV, 328 p. (1952; Zbl 0047.41402)], on the one hand, and its far-reaching conclusions on the other. Thus the methodological (and didactical) conception of the presentation is a mainly formal and deductive one, where the focus is much more on the algebraic framework than on geometric-topological intuition. As the author already points out in the preface, the text does not contain the proofs of all the theorems, and the results are not always stated in their greatest possible generality, essentially with regard to the intended character of the book as a pragmatic guide to the subject. More precisely, while most proofs are given (or at least sketched), others have been left as exercises for the reader, and some of the more involved, particularly long, or extremely technical proofs have been omitted at all, thus maintaining the predominant focus on panoramic overviews and concrete applications throughout the text.
As to the contents, this terse guide to algebraic topology is organized in seven chapters and three appendices. Assuming the basic knowledge of general topology and abstract algebra from undergraduate courses, Chapter 1 very briefly introduces the notions of homotopy for pairs of topological spaces, homotopy equivalence, deformation retract and contractibility, respectively. Chapter 2 discusses fundamental groups, covering spaces, van Kampen’s theorem (without proof), and some purely algebraic applications of this homotopical framework to the study of free abstract groups. Chapter 3 presents the Eilenberg-Steenrod axioms for a generalized homology theory for pairs of topological spaces, then proceeds by deriving various standard consequences from these axioms, and finally turns to the Eilenberg-Steenrod axioms for cohomology theory. Chapter 4 specializes to ordinary homology theory, i.e., a general homology theory satisfying the so-called dimension axiom and having the integers as coefficient group. Continuing to proceed axiomatically, the author derives many classical results and applications in this context, including the computation of homology groups of spheres, Brouwer’s fixed-point theorem, and the theorem on the invariance of domain. Furthermore, CW-complexes and their cellular homology as well as the ordinary homology of real and complex projective spaces are touched upon in this fourth chapter. Chapter 5 develops the classical singular homology theory as a concrete example of an abstract ordinary homology theory, and presents many of its main properties. In this context, singular homology with arbitrary coefficients and the universal coefficient theorem, the Künneth formula, singular cohomology theory, cup and cap products, and the famous Borsuk-Ulam theorem are explained as well. Chapter 6 treats topological manifolds, the concept of orientability, examples of orientable and non-orientable manifolds, respectively, then states the duality theorems of Poincaré and Lefschetz for compact oriented manifolds, and finally gives some of the important classical applications of these central theorems. The main text ends with Chapter 7, where a short introduction to higher homotopy groups is provided, together with an outlook concerning some of the great classical theorems in homotopy theory due to Hurewicz, Freudenthal, Serre, and others.
The three appendices briefly compile some of the purely algebraic tools used in the course of the text. Appendix A presents a few basic facts from elementary homological algebra, including exact sequences of module homomorphisms, chain complexes and long homology sequences, tensor products, the functors Hom, Ext, Tor and the standard long exact sequences concerning the latter. Appendix B states the basic classification theorems for non-singular bilinear forms on Abelian groups or finite-dimensional real vector spaces, respectively, whereas Appendix C recalls the basic terminology from the theory of categories and functors.
Each chapter of the book ends with a list of exercises related to the respective material, where most of the problems ask the reader to give proofs of results stated in the text. The formulation of these exercises it utmost terse, and hints for solution are given nowhere.
This, together with the very concise style of presention of the basics of abstract algebraic topology, requires quite a bit of mathematical maturity from the reader, who certainly will need some additional reading for complete understanding. On the other hand, the conciseness of the book under review is also its outstanding strength, with a view toward seasoned, mathematically experienced readers, as it provides a very rapid introduction to the subject, and a great incentive for further, more detailed and advanced study likewise.
All in all, the present book is certainly a highly useful and valuable companion for a first-year graduate course in algebraic topology, as well for ambitious students as for instructors.

55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
55N40 Axioms for homology theory and uniqueness theorems in algebraic topology
55N10 Singular homology and cohomology theory
55Q05 Homotopy groups, general; sets of homotopy classes
55U25 Homology of a product, Künneth formula
55U30 Duality in applied homological algebra and category theory (aspects of algebraic topology)
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