Introduction to Topological Manifolds

This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of di?erential geometry, algebraic topology, and related ?elds. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. Here at the University of Washington, for example, this text is used for the ?rst third of a year-long course on the geometry and topology of manifolds; the remaining two-thirds focuses on smooth manifolds. Therearemanysuperbtextsongeneralandalgebraictopologyavailable. Why add another one to the catalog? The answer lies in my particular visionofgraduateeducation—itismy(admittedlybiased)beliefthatevery serious student of mathematics needs to know manifolds intimately, in the same way that most students come to know the integers, the real numbers, Euclidean spaces, groups, rings, and ?elds. Manifolds play a role in nearly every major branch of mathematics (as I illustrate in Chapter 1), and specialists in many ?elds ?nd themselves using concepts and terminology fromtopologyandmanifoldtheoryonadailybasis. Manifoldsarethuspart of the basic vocabulary of mathematics, and need to be part of the basic graduate education. The ?rst steps must be topological, and are embodied in this book; in most cases, they should be complemented by material on smooth manifolds, vector ?elds, di?erential forms, and the like. (After all, few of the really interesting applications of manifold theory are possible without using tools from calculus.

“Each chapter contains relatively easy exercises and ends with a section containing problems … . This clearly written book represents, to a good approximation … . 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.” (Mircea Craioveanu, zbMATH 0956.57001, 2022)

"This book is an introduction to manifolds on the beginning graduate level. It provides a readable text allowing every mathematics student to get a good knowledge of manifolds in the same way that most students come to know real numbers, Euclidean spaces, groups, etc. It starts by showing the role manifolds play in nearly every major branch of mathematics.

The book has 13 chapters and can be divided into five major sections. The first section, Chapters 2 through 4, is a brief and sufficient introduction to the ideas of general topology: topological spaces, their subspaces, products and quotients, connectedness and compactness.

The second section, Chapters 5 and 6, explores in detail the main examples that motivate the rest of the theory: simplicial complexes, 1- and 2-manifolds. It introduces simplicial complexes in both ways---first concretely, in Euclidean space, and then abstractly, as collections of finite vertex sets. Then it gives classification theorems for 1-manifolds and compact surfaces, essentially following the treatment in W. Massey's \ref[ Algebraic topology: an introduction, Reprint of the 1967 edition, Springer, New York, 1977; MR0448331 (56 \#6638)].

The third section (the core of the book), Chapters 7--10, gives a complete treatment of the fundamental group, including a brief introduction to group theory (free products, free groups, presentations of groups, free abelian groups), as well as the statement and proof of the Seifert-Van Kampen theorem.

The fourth major section consists of Chapters 11 and 12, on covering spaces, including proofs that every manifold has a universal covering and that the universal covering space covers every other covering space, as well as quotients by free proper actions of discrete groups.

The last Chapter 13 covers homology theory, including homotopy invariance and the Mayer-Vietoris theorem.

The book gives an ample opportunity to the reader to learn the subject by working out a large number of examples, exercises and problems. The latter are collected at the end of each chapter."  (B.N. Apanasov, Mathematical Reviews)