Basic algebraic topology (Q2840667)

This comprehensive textbook grew out of basic courses in algebraic topology that the author has taught since the late 1980s at the Indian Institute of Technology in Mumbai. As the author points out in the preface, the text is intended for a two-semester first course on the subject, thereby assuming that the readers are familiar with the fundamental concepts of real analysis, general topology, abstract algebra, and differential topology. However, the book contains much more material than an instructor could cover in two semesters, which leaves much freedom concerning the choice of topics to be taught (or studied) from later, more advanced chapters. Also, the author acknowledges that he has been strongly influenced by \textit{Edwin H. Spanier's} classic book ``Algebraic Topology'' [McGraw-Hill Series in Higher Mathematics. New York etc.: McGraw-Hill Book Company. XIV, 528 p. (1966; Zbl 0145.43303)], the probably first comprehensive treatment of the subject in the textbook literature. In fact, the present book may be considered as a didactically elaborated, appropriately up-dated and more student-friendly version of Edwin H. Spanier's very abstract, nearly encyclopedic standard text from almost fifty years ago.NEWLINENEWLINE As for the precise contents, the book comprises thirteen chapters, each of which contains several thematic sections.NEWLINENEWLINE Chapter 1 is of introductory nature and discusses the general question of what algebraic topology is all about, mainly through examples from elementary homotopy theory. Furthermore, a number of basic topological concepts and constructions is discussed, including fundamental groups, relative homotopy, mapping cones and mapping cylinders, cofibrations, fibrations, and a brief introduction to categories and functors.NEWLINENEWLINE Chapter 2 treats convex polytopes, cell complexes and their homotopical aspects, abstract simplicial complexes and their geometric realization, the method of barycentric subdivision, simplicial approximations, and finally links and stars of simplicial complexes.NEWLINENEWLINE Chapter 3 presents the standard material on covering spaces and fundamental groups, along with the study of group actions and Grothendieck's concept of \(G\)-coverings in this context. The Seifert-van Kampen theorem and some concrete applications conclude this chapter.NEWLINENEWLINE The discussion of homology theory starts in Chapter 4, where the construction of singular homology groups and their properties is the central topic. In the sequel, some other homology theories are also introduced, including smooth singular homology for smooth manifolds, simplicial homology as well as (cellular) CW-homology for CW-complexes.NEWLINENEWLINE Apart from the instructive standard applications such as Brouwer's and Lefschetz's fixed point theorems, the Jordan-Brouwer separation theorem and Brower's theorem on the invariance of domains, Hurewicz's isomorphism theorem relating the fundamental group and the first singular homology group is also presented.NEWLINENEWLINE Chapter 5 is devoted to topological manifolds and their triangulation properties. In this context, the topological classification of surfaces (via their triangulations) is derived, and a first introduction to topological vector bundles is provided at the end of this chapter.NEWLINENEWLINE More algebraic tools for homology theory are developed in Chapter 6, with particular emphasis on the method of acyclic models, the Tor functor and homology with coefficients, the universal coefficient theorems for homology, the Eilenberg-Zilber theorem, and the Künneth formula.NEWLINENEWLINE Chapter 7 turns to cohomology theory for topological spaces, culminating in an explanation of products in the cohomology ring, Steenrod's squaring operations, Adem's relations, and a number of explicit cohomological computations. The return to the study of topological manifolds takes place in Chapter 8, where the notion of orientability is analyzed and the topological duality theorems (à la Alexander, Lefschetz and Poincaré) are proven. Also, various applications of the duality theorems are included, and a proof of de Rham's theorem for smooth manifolds is given without using sheaf cohomology.NEWLINENEWLINE Chapter 9 deals with sheaves and their cohomology, thereby developing the necessary concepts and techniques from homological algebra in a self-contained manner. Within this framework, Čech cohomology for sheaves is discussed as well, and the standard proof of de Rham's theorem is given as an application of the latter.NEWLINENEWLINE Chapter 10 is what the author calls the heart of the book. Being titled ``Homotopy Theory'', this chapter touches upon various topics related to higher homotopy groups and their basic properties. The reader meets here Hurewicz's general isomorphism theorems, the Whitehead theorem, the basics of obstruction theory, the results of Eilenberg and Hopf-Whitney on the homotopy classification of maps, Eilenberg-MacLane spaces and their properties, the Moore-Postnikov decomposition, homotopy groups of classical Lie groups and their quotients, homology with local coefficients, and many other important results in higher homotopy theory.NEWLINENEWLINE Chapter 11 returns to the study of homology groups. More precisely, the central theme is here the homology of fibre spaces, which starts with some generalities about fibrations, presents then the Thom isomorphism theorem and the Gysin exact homology sequence for \(n\)-spherical fibrations, the Leray-Hirsch theorem, Freudenthal's homotopy suspension theorem, and concludes with computing the cohomology algebra of some of the classical groups. Chapter 12 provides a brief introduction to characteristic classes of vector bundles, including the Euler class, the Stiefel-Whitney classes, and the Chern classes. At the end, complex vector bundles and their Pontryagin classes are discussed, together with some applications to oriented cobordism theory (à la R. Thom).NEWLINENEWLINE Finally, Chapter 13 introduces the basics of the theory of spectral sequences and its applications in algebraic topology. After the brief discussion of some generalities, the Leray-Serre spectral sequence of a fibration is analyzed in greater detail, along with its immediate applications to generalized homology sequences (à la Borel, Gysin, Serre and Wang). Cohomology spectral sequences, Serre classes of Abelian groups, and the generalization of several homotopy-theoretic results of Chapter 10 are further applications of spectral sequences presented in this concluding chapter.NEWLINENEWLINE Each of the first eleven chapters ends with a section titled ``Miscellaneous Exercises''. The working problems listed here are carefully selected and meant as part of the main text, thereby providing a wealth of related and additional material for the reader's active, independent work of study. Further exercises are found at the end of each single section, where they directly refer to the respective topic discussed there, and a large number of illustrating, highly instructive examples and remarks is scattered throughout the entire text, which effectively helps the reader grasp the abundance of abstract concepts, methods and techniques in modern algebraic topology. Moreover, there is a hint/solution manual at the end of the book for some selected exercises, and a ``sectionwise dependence tree'' may help both teachers and students to make their own plan for using this volume.NEWLINENEWLINE Overall, this excellent textbook provides a very comprehensive, utmost versatile and profound introduction to the principles of algebraic topology. The exposition stands out by its high degree of lucidity, mathematical rigor, didactic mastery, and self-containedness with regard to seasoned students. In fact, this textbook offers much more (advanced) material than most other primers in the field do, and therefore it certainly deserves to become one of great standard texts in algebraic topology for teachers, students, and young researchers in the future.