Invitation to algebraic topology. Vol. I. Homology (Q2876103)

The two-volume textbook [Invitation to algebraic topology. Vol. II. Cohomology, manifolds. (Invitation à la topologie algébrique. Tome II. Cohomologie, variétés.), ibid. 298 p. (2014; Zbl 1297.55002); ISBN 978-2-36493-127-5/pbk] by the authors provides an introduction to some of the cornerstones of the subject with a view toward upper-level undergraduates and graduate students. Without striving for encyclopedic completeness, the authors focus on three major aspects which are treated in the three parts of the treatise: homology (Part I), cohomology (Part II), and manifolds (Part III). In this approach, both the discussion and the use of homotopy theory are reduced to an absolute minimum, whereas applications of (co-)homological methods to the study of topological manifolds play a predominant role. As for an invitation to algebraic topology, this particular didactic approach is certainly very suitable and useful, especially with regard to novices in the field.NEWLINENEWLINE The book under review is the first volume of the text. It contains the first fourteen chapters, including Part I titled ``Homology''. After a very appealing historical introduction, Chapter 1 briefly compiles the necessary background material from general topology such as homotopies, the fundamental group, covering spaces, topological manifolds, topological surfaces, and projective spaces.NEWLINENEWLINE Chapter 2 is still of preliminary nature and presents some complements from algebra, including a little module theory, tensor products, categories, and functors as used in the sequel.NEWLINENEWLINE As for the general prerequisites, the reader is assumed to have acquired a profound basic knowledge of both general topology and abstract algebra, and therefore the first two chapters are merely short reminders for the convenience of the reader. In this respect, the authors' earlier textbook [An invitation to algebra. Theory of groups, rings, fields and modules. (Invitation à l'algèbre. Théorie des groupes, des anneaux, des corps et des modules.) Mathématiques. L3-Master-CAPES-Agreg. Toulouse: Cépaduès-Éditions. xiv, 394 p. (2008; Zbl 1163.13001); ISBN 978-2-85428-740-0/pbk] may serve as an excellent reference for the algebraic background of the current text.NEWLINENEWLINE The following Chapters 3--14 constitute Part I of the entire book and deal with homology theory. Chapter 3 introduces simplicial complexes, while Chapter 4 explains the concept of simplicial homology. Algebraic chain complexes and the long homology sequence are discussed in Chapter 5, and the Mayer-Vietoris sequence as well as the long homology sequences for pairs and triples of simplicial complexes, together with the related excision theorem, are derived in Chapter 6. In the sequel, (relative) singular homology is the topic of Chapter 7, the homotopy invariance of singular homology is shown in Chapter 8, and various methods for calculating singular homology groups (e.g., excision, Mayer-Vietoris sequences, and attaching cells) are demonstrated in Chapter 9. Applications of homology theory such as the theorems of Brouwer and Jordan, the degrees of maps between spheres, and the invariance of dimension for topological manifolds are presented in Chapter 10, while Chapter 11 treats the homology of polyhedra, simplicial approximations, and the comparison between simplicial and singular homology. Chapter 12 discusses further constructions concerning abstract chain complexes, including their tensor products, resolutions, the Künneth theorem, and the extension of coefficients. This is continued in Chapter 13, where homology with coefficients is described for both simplicial and singular homology. The universal coefficient theorem, the Euler characteristic, and the Lefschetz number are further topics touched upon in this chapter. Part I of the book ends with Chapter 14, which discusses the homology of product spaces, first in the absolute case and subsequently in the relative case.NEWLINENEWLINE In a supplement to Part I, a few additional remarks to the topics treated in some of the chapters are made.NEWLINENEWLINE Each of the chapters ends with an extra section providing a selection of related exercises, and the whole text is enriched by numerous instructive examples and elucidating remarks. The representation of the material stands out by a high degree of clarity, detailedness, and mathematical rigor.NEWLINENEWLINE Part 2 and Part 3 of the book form the second volume, which will be reviewed immediately afterwards [the authors, loc. cit.].