Introduction to Topological ManifoldsSpringer Science & Business Media, 2000 - 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. |
Table des matières
Introduction | 1 |
Topological Spaces 17 | 16 |
New Spaces from Old | 39 |
Connectedness and Compactness | 65 |
Simplicial Complexes | 90 |
Curves and Surfaces | 117 |
Problems | 146 |
Circles and Spheres | 178 |
Some Group Theory | 193 |
The SeifertVan Kampen Theorem | 209 |
Covering Spaces | 233 |
Covering Maps and the Fundamental Group | 239 |
Classification of Coverings | 257 |
Homology | 291 |
Review of Prerequisites | 336 |
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Expressions et termes fréquents
algebraic base point basis boundary called chapter characteristic property circle component compute connected sum contained continuous map Corollary covering map covering space Cp(X cyclic group define definition deformation retraction denote dimension disk element equivalence class equivalence relation Euler characteristic evenly covered Example Exercise exists Figure finite free abelian group free group free product function fundamental group given Hausdorff space homeomorphic homology groups induced infinite cyclic injective integer intersection inverse isomorphism Lemma lifting property loop manifold map f metric space morphisms n-manifold neighborhood nonempty open cover open sets open subset orientation pair path class path connected path homotopic polygonal Proof Proposition prove quotient map quotient space second countable sequence simplex simplicial complex simply connected singular sphere subgroup subspace topology Suppose surjective theorem theory topological space triangulation unique vector vertex vertices