Introduction to Topological ManifoldsSpringer Science & Business Media, 6 avr. 2006 - 392 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 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. |
Table des matières
| 1 | |
Topological Spaces 17 | 16 |
New Spaces from Old | 39 |
Connectedness and Compactness | 65 |
Simplicial Complexes 91 | 90 |
Curves and Surfaces | 117 |
Homotopy and the Fundamental Group | 147 |
Circles and Spheres 179 | 178 |
Some Group Theory | 193 |
The SeifertVan Kampen Theorem | 209 |
Covering Spaces | 233 |
Classification of Coverings | 257 |
Homology | 291 |
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Expressions et termes fréquents
abelian group algebraic ball base point basis boundary called chapter characteristic property circle compact surfaces compute connected sum contained continuous map Corollary covering map covering space covering transformation define definition deformation retraction denote dimension disk edge path element equivalence class equivalence relation Euler characteristic evenly covered Example Exercise exists fiber Figure finite free abelian free group free product function fundamental group given Hausdorff space homeomorphic homology groups homotopy equivalent identity map induced infinite cyclic injective integer intersection inverse isomorphism lifting property loop manifold map f metric space morphisms n-manifold neighborhood nonempty open cover open sets open subset orientation pair path class path homotopic polygonal projective planes Proof Proposition prove quotient map quotient space second countable sequence simplex simplicial complex simply connected sphere subgroup subspace topology Suppose surjective theorem theory topological space triangulation unique universal covering vector vertex vertices