Invitation to algebraic topology. Vol. II. Cohomology, manifolds
type: Item
instance of: scholarly article
zbMATH Open document ID: 1297.55002
author: Alain Jeanneret
author: Daniel Lines
publication date: 2014-08-15T00:00:00Z
Mathematics Subject Classification ID: 55-01
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 57N16
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 55N45
Mathematics Subject Classification ID: 57Q15
zbMATH DE Number: 6330953
zbMATH Keywords: textbook (algebraic topology)
zbMATH Keywords: cohomology theory
zbMATH Keywords: products and intersections
zbMATH Keywords: topological manifolds
zbMATH Keywords: triangulations
zbMATH Keywords: orientable manifolds
zbMATH Keywords: cohomological duality theorems
MaRDI profile type: Publication
title: Invitation to algebraic topology. Vol. II. Cohomology, manifolds
review text: 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, differen…
reviewed by: Werner Kleinert
description: scientific article; zbMATH DE number 6330953