Invitation to algebraic topology. Vol. II. Cohomology, manifolds
This is the second volume of the authors' introductory textbook on some basic principles of algebraic topology. While the first volume with the subtitle [Invitation to algebraic topology. Vol. I. Homology. (Invitation à la topologie algébrique. Tome I. Homologie.) ibid. 297 p. (2014; Zbl 1297.55001); ISBN 978-2-36493-126-8/pbk] is devoted to simplicial and singular homology, this second volume comprises Part II and Part III of the entire text and switches over to some more advanced topics.NEWLINENEWLINE Part II is titled ``Cohomology'' and contains the two Chapters 15 and 16. The basic definitions and examples of both the singular and the simplicial cohomology theory are developed in Chapter 15. This is done after an introduction to abstract cochain complexes and cohomology groups, long cohomology sequences, and universal coefficients via the functors Ext and Tor in the first two sections of this chapter, and two theorems of Hopf on maps from simplicial complexes to spheres serve as instructive applications of cohomological obstruction theory to geometric problems. Chapter 16 discusses the standard product operations in cohomology, notably the cross product, the cup product, the cap product, and the slant product. These cohomological products are mainly used later in the study of the cohomology of manifolds, but some more elementary applications of them toward the computation of the low cohomology groups of projective spaces are already exhibited at this stage.NEWLINENEWLINE Part III turns to the topic of homology and cohomology of topological and differentiable manifolds. Chapter 17 describes the fundamentals of topological, differentiable and triangulable manifolds, with special emphasis on manifolds with boundary, on the one hand, and the finiteness properties of the homology of compact manifolds on the other. Chapter 18 gives a detailed presentation of the concept of orientation for manifolds, together with its different interpretations for topological, differentiable or triangulable manifolds, respectively. Chapter 19 is devoted to the celebrated duality theorems of Poincaré, Alexander, and Lefschetz for various types of manifolds, thereby using the full cohomological framework as developed in Part II. In this context, the bilinear intersection form on the cohomology on an oriented, compact manifold of even dimension is also explained.NEWLINENEWLINE In the final Chapter 20, the authors survey a number of spectacular results of the past fifty years in the study of manifolds, including PL-manifolds and the related ``Hauptvermutung'', theorems in low-dimensional topology, the Poincaré conjecture, non-combinatorial triangulations of manifolds, bordism groups and rings, embeddings of manifolds, and other highlights in modern algebraic topology. The discussion here is informal, without any proofs, but utmost enlightening and appetizing.NEWLINENEWLINE There are two appendices at the end of the book, which both collect some facts on direct limits (Appendix A) and bilinear forms on modules (Appendix B), respectively. These technical tools are used in the course of Part III particularly in Chapter 19.NEWLINENEWLINE As in the first volume, each chapter contains an extra section with related, carefully selected exercises, and the entire text is again interspersed with a wealth of illustrating examples and remarks. Also, each of the two main parts of the book is supplemented by a number of additional notices and hints with respect to the topics discussed in the single chapters.NEWLINENEWLINE Again, as in the first volume, the representation of the material is utmost lucid, detailed, rigorous and inspiring. All together, the two volumes provide a very profound and appealing introduction to the (co-)homological methods in the algebraic topology of manifolds.
scientific article; zbMATH DE number 6330953