# Introduction to Topological Manifolds

Springer Science & Business Media, Jan 1, 2000 - Mathematics - 385 pages
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 differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. A course on manifolds differs from most other introductory mathematics graduate courses in that the subject matter is often completely unfamiliar. Unlike algebra and analysis, which all math majors see as undergraduates, manifolds enter the curriculum much later. It is even possible to get through an entire undergraduate mathematics education without ever hearing the word "manifold." Yet manifolds are part of the basic vocabulary of modern mathematics, and students need to know them as intimately as they know the integers, the real numbers, Euclidean spaces, groups, rings, and fields. In his beautifully-conceived Introduction, the author motivates the technical developments to follow by explaining some of the roles manifolds play in diverse branches of mathematics and physics. Then he goes on to introduce the basics of general topology and continues with the fundamental group, covering spaces, and elementary homology theory. Manifolds are introduced early and used as the main examples throughout. John M. Lee is currently Professor of Mathematics at the University of Washington in Seattle. In addition to pursuing research in differential geometry and partial differential equations, he has been teaching undergraduate and graduate courses on manifolds at U.W. and Harvard University for more than fifteen years.

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### Contents

 Introduction I 1 Topological Spaces 17 New Spaces from Old 39 Connectedness and Compactness 65 Simplicial Complexes 01 91 Curves and Surfaces 117 Homotopy and the Fundamental Group 147 Circles and Spheres 179
 The SeifertVan Kampen Theorem 209 Covering Spaces 233 Classification of Coverings 257 Homology 292 Review of Prerequisites 337 Metric Spaces 347 References 359 Copyright

 g Some Group Theory 103 194