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Introduction to topological manifolds. (English) Zbl 0956.57001

Graduate Texts in Mathematics. 202. New York, NY: Springer. xvii, 385 p. (2000).
Manifolds play a role in many major branches of mathematics. Thus they are part of the basic vocabulary of mathematics, and need to be part of the basic graduate education. This book is an introduction to topological manifolds at beginning graduate level. It represents an expanded version of notes designated for a one-quarter graduate course and contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology and related fields. As the author himself remarks, its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition.
Let us give a short description of the contents. The book starts with an introductory chapter, beginning with a non-rigorous definition of manifolds, going on with an informal exploration of some examples, and ending with a consideration of where and why they arise in various branches of mathematics. The second through the fourth chapters are a brief and highly selected introduction to the ideas of general topology: topological spaces, their subspaces, products, quotients, connectedness, compactness. Topological manifolds are the main examples and are emphasized throughout. The author carefully develops the machinery that will be used later, such as topological projections (identification maps), local path connectedness and locally compact Hausdorff spaces. The fifth chapter introduces simplicial complexes in two ways – first concretely, as locally finite collections of simplices in Euclidean space that intersect nicely, and then abstractly, as collections of finite vertex sets. Then the author applies these ideas to manifolds, giving a complete proof that every 1-manifold is triangulable and a brief sketch of the proof for 2-manifolds. Also he explores two combinatorial properties of simplicial complexes that are important in the study of manifolds: the orientation of a complex and the Euler characteristic. Using the triangulability theorem, the sixth chapter begins by proving a classification theorem for 1-manifolds. After discussing the basic examples of surfaces, the author gives a complete proof of the classification theorem for closed surfaces, essentially following the treatment in W. S. Massey [Algebraic topology, Grad. Texts Math. 56 (1981; Zbl 0457.55001)]. The seventh through the tenth chapters are the core of the book, offering a fairly complete and traditional treatment of the fundamental group. The seventh chapter introduces the definitions and proves the topological and homotopy invariance of the fundamental group. The eighth chapter gives a detailed proof that the fundamental group of the circle is infinite cyclic. Also, the author computes the fundamental group of higher-dimensional spheres and product spheres and he shows that the fundamental group of any manifold is countable. The ninth chapter is a brief digression into some topics of group theory: free products, free groups, presentation of groups and free abelian groups. The tenth chapter gives the proof of a rather general version of the Seifert-Van Kampen theorem, and derives several applications of it, including computation of the fundamental group of graphs and of all closed surfaces. The eleventh chapter defines covering spaces, gives a few examples and develops the theory of the covering group. The twelfth chapter proves the classification theorem for all coverings up to isomorphism. It is also proven that every manifold has a universal covering and shown how to construct coverings as quotients of a given space by certain group actions. The author illustrates the techniques by determining the universal covering spaces of all closed surfaces, classifying all the coverings of the torus and the lens spaces, and proving that surfaces of higher genus are covered by the hyperbolic disk. The thirteenth chapter starts by defining the singular homology groups of a topological space. Then a few essential properties, including homotopy invariance, the Mayer-Vietoris theorem as well as the relationship between the first homology group and the fundamental group, are proven. The homology groups of most of the spaces that have been studied before are computed. Then the author describes some applications of homology: the topological invariance of the dimension of a manifold, the existence of vector fields on spheres, and – using simplicial homology – the topological invariance of the Euler characteristic of a polyhedron. The chapter ends with a brief introduction to cohomology. There is an appendix, reviewing the prerequisites on some fundamental aspects of set theory, metric spaces and group theory that are used throughout, together with a representative collection of exercises. Each chapter contains relatively easy exercises and ends with a section containing problems that are, in general, harder or longer than the exercises.
This clearly written book represents, to a good approximation, the author’s conception of the ideal amount of topological knowledge that should be possessed by beginning graduate students who are planning to go on to study smooth manifolds and differential geometry. It is an excellent source for teaching a course precursory to algebraic topology, smooth manifolds and differential geometry and contains enough material for a one-year course. In any case, this very nicely conceived book is a worthwhile addition to any mathematical library.

MSC:

57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
55N10 Singular homology and cohomology theory
57N65 Algebraic topology of manifolds
57M05 Fundamental group, presentations, free differential calculus
57M10 Covering spaces and low-dimensional topology
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57Q05 General topology of complexes
55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
55Q05 Homotopy groups, general; sets of homotopy classes

Citations:

Zbl 0457.55001
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