Introduction to Topological Manifolds

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Springer Science & Business Media, Jan 1, 2000 - Mathematics - 385 pages
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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

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