Algebraic topology. An introduction (Q2901225)

This graduate textbook provides a comprehensive introduction to some classical topics in algebraic topology, with a particular emphasis on simplicial and singular (co)homology. Certainly, there is a large number of excellent standard textbooks on the subject, and the contents of the present book cover much of the traditional material presented in those primers on algebraic topology. However, the book under review is much more than just another typical introduction to the subject, since it offers some remarkable, distinctive features. First of all, the authors do not follow the usual pattern of starting an introduction to algebraic topology by explaining coverings, homotopy, and fundamental groups. Secondly, with the main focus on the classical algebraic invariants based on simplicial and singular (co)homology theory from the very beginning on, they have chosen the very geometric viewpoint of singular bordism theory and pseudomanifolds in the customary transition from simplicial to singular homology, which can be found in just a few introductory monographs otherwise. As for the precise contents, this textbook comprises fifteen chapters, each of which is composed of several sections.NEWLINENEWLINE In a brief introduction to what follows, novices are provided with a naive approach to the idea of homology, together with an apt historical sketch of its development from Riemann to Eilenberg and Steenrod. Chapter 1 presents an overview of basic topological concepts, including quotient and attaching spaces, surfaces and manifolds, topological groups and group actions, and some elementary notions of homotopy theory. Chapter 2 gives a first introduction to polyhedral topology, simplicial complexes, simplicial maps, simplicial approximations, barycentric subdivisions, and triangulations. The generalization to abstract simplicial complexes, triangulations of quotients and orbit spaces, and the concept of CW-complexes are expounded in Chapter 3, while the necessary basic notions from homological algebra are made available in the following Chapter 4. Simplicial (co)homology is introduced and developed in Chapter 5, with the notions of cup-product and cap-product concluding the discussion of this part. Chapter 6 treats a very general framework for describing homology classes, namely the theory of pseudomanifolds and \(n\)-polyhedra due to L. E. J. Brouwer (1910).NEWLINENEWLINE Apart from the basic geometrical constructions, the relationship between orientation and simplicial (co)homology is discussed in great detail. Chapter 7 is devoted to the proofs of the classical duality theorems of Poincaré, Alexander and Lefschetz for homology manifolds. The study of the elementary properties of singular bordism theory is carried out in Chapter 8, where polyhedral topology serves as the fundamental framework. Then, in Chapter 9, a suitable theory of pseudobordism is developed for pseudomanifolds. This approach to homology culminates in the proof of the equivalence between pseudobordism groups and simplicial homology groups for polyhedra (or CW-complexes, respectively).NEWLINENEWLINE In this context, a brief overview of a purely geometrical definition of simplicial cohomology is given as well. Chapter 10 turns to singular (co)homology theory, thereby treating both the usual algebraic and the pseudobordism approach to singular homology, proving the equivalence of the two approaches, and describing the Eilenberg-Steenrod axioms for general (co)homology theories functorially. The first classical applications of the homology theories, as developed so far, are given in Chapter 11, where the Brouwer fixed-point theorem, the Hopf trace theorem, and the Lefschetz-Hopf fixed-point theorem are dealt with. Chapter 12 discusses further applications, with the Jordan-Brouwer separation theorem and the Borsuk separation criterion being the central topics here, whereas Chapter 13 is concerned with the theory of degree for topological maps, together with its classical applications.NEWLINENEWLINE Chapter 14 turns then to the rudiments of homotopy theory. The definition of (higher) homotopy groups, the Seifert-Van Kampen theorem, and the Hurewicz homomorphism relating homology and homotopy are the main contents of this section. Finally, Chapter 15 treats the homotopy classification of lens spaces as an instructive example of the significance of the algebraic invariants introduced in the course of the present book.NEWLINENEWLINE In order to ease the exposition, the authors have collected some results and proofs in a series of appendices at the end of the volume. Each chapter comes with its own list of related exercises and problems. Many of the problems are sketches of further relevant topics in algebraic topology. In this way, a wealth of additional material enhances the main text. Also, the solutions to numerous marked exercises and problems must be considered as an integral part of the respective chapter in the main text. Altogether, there is an utmost ample variety of exercises and problems for the reader's independent work, further study, and self-control.NEWLINENEWLINE No doubt, the current book gives a highly individual, valuable, lucid, detailed, abundant and versatile introduction to algebraic topology, with many instructive pictures illustrating the abstract material, numerous concrete examples, and a just as huge number of clarifying remarks throughout the text. It is absolutely evident that the authors have composed this primer with great enthusiasm, expository mastery, didactic experience, and empathy for beginners in the field.NEWLINENEWLINE Therefore it is fair to say that the book under review is a highly useful enhancement of the existing textbook literature in the field.