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Lee J.M. — Introduction to Topological Manifolds
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Название: Introduction to Topological ManifoldsАвтор: Lee J.M. Аннотация: 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.
Язык: Рубрика: Математика /Геометрия и топология /Общая топология /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2000Количество страниц: 385Добавлена в каталог: 19.04.2005Операции: Положить на полку |
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Предметный указатель
Poincare, homomorphism 305 Point 18 Point at infinity, ideal 12 Pointed, homotopy category 173 Pointed, topological category 171 Pointed, topological space 171 Polar coordinates 3 Pole, north 3 45 Pole, south 45 Polygon, geodesic 273 Polygon, regular geodesic 274 Polygonal presentation 130 Polygonal presentation, geometric realization 131 Polygonal presentation, topologically equivalent 133 Polygonal region 123 Polyhedron 100 Polyhedron is Hausdorff 114 Polyhedron is locally path connected 114 Polyhedron, Euclidean 94 Polyhedron, fundamental group 230 Polyhedron, homology 334 Polynomial 20 Position, general 92 Positivity of metric 348 Positivity of norm 89 Power map 191 235 Power set 339 Power set, axiom 339 Power set, partial ordering 341 Precompact 82 Presentation and Seifert — Van Kampen theorem 211 Presentation of a group 201 202 Presentation, polygonal 130 Presentation, polygonal, geometric realization 131 Presentation, polygonal, topologically equivalent 133 Presentation, standard 133 137 Presentation, surface 132 Presentation, surface and fundamental group 217 Presentation, surface, classification 137 Product in a category 174 Product in a category, uniqueness 174 Product of closed maps 62 Product of compact spaces 74 Product of covering maps 253 Product of locally compact, Hausdorff spaces 83 Product of manifolds 51 Product of open maps 62 Product of path classes 153 Product of paths 152 Product of quotient maps 86 Product of topological groups 59 Product of words 194 Product, cartesian 340 346 Product, Cartesian, finite 346 Product, Cartesian, infinite 346 Product, direct 353 Product, dot 347 Product, free 195 Product, map 50 Product, open sets 48 Product, space 48 Product, space, connectedness 67 Product, space, fundamental group 189 Product, space, Hausdorff 50 Product, space, second countable 50 Product, topology 48 Product, topology on 48 Product, topology, associativity 50 Product, topology, basis 48 50 Product, topology, characteristic property 49 Product, topology, infinite 49 177 Product, topology, uniqueness 49 Projection from a Cartesian product 346 Projection from a product space 50 Projection from a product space is a quotient map 62 Projection in a category 174 Projection onto a quotient group 355 Projection onto a quotient space 52 Projection, stereographic 45 187 Projection, stereographic and one-point compactification 89 Projective plane 119 Projective plane, covering 253 Projective plane, Euler characteristic 143 Projective plane, fundamental group 220 247 Projective plane, presentation 133 Projective plane, quotient of disk 122 Projective plane, quotient of sphere 121 Projective plane, quotient of square 122 Projective space as orbit space 61 Projective space is a manifold 62 Projective space, complex 12 62 Projective space, covering 253 Projective space, homology 334 Projective space, real 55 Projective transformation 12 Proper, face 93 Proper, group action 266 Proper, group action on locally compact Hausdorff space 267 Proper, group action, quotient 268 Proper, local homeomorphism 253 Proper, map 84 Proper, map is closed 84 Proper, subset 338 Properly discontinuous group action 268 Property, topological 4 Pseudomanifold 334 Quantum field theory 14 Quotient by free proper group action 268 Quotient by group action 61 Quotient of a compact space 74 Quotient of a manifold 269 Quotient of a topological group 63 Quotient, descending to 56 57 Quotient, group 355 Quotient, map 52 Quotient, map, characterization 57 Quotient, map, composition 53 Quotient, map, exponential 61 235 Quotient, map, restriction 53 Quotient, passing to 56 57 Quotient, second countable 54 Quotient, space 52 Quotient, space, connectedness 67 Quotient, space, uniqueness 57 Quotient, topology 52 Quotient, topology, characteristic property 56 Rado, Tibor 104 RANGE 341 342 Rank of a finitely generated abelian group 206 Rank of a free abelian group 205 Rational function 20 Rational numbers 344 Real line 2 Real numbers 343 Real numbers, uniqueness 343 Real projective space 55 Real projective space is a manifold 62 Real vector spaces, category of 171 Realization, geometric 97 99 Realization, geometric, of a polygonal presentation 131 Reduced edge path 101 Reduced word 195 Reduction algorithm 196 Reduction, elementary 195 Reflecting 134 Reflection map 321 Reflexive 340 Region, polygonal 123 Regular Euclidean ball 83 Regular geodesic polygon 274 Regular hyperbolic neighborhood 278 Regular point, of vector field 192 Relabeling 133 Relation 340 Relation of a presentation 202 Relation, equivalence 52 340 Relation, equivalence, generated by a relation 340 Relative homotopy 151 Relative topology 40 Relatively compact 82 Relativity, general 14 Relator 201 Reparametrization 151 Restriction 342 Restriction of quotient map 53 Restriction, continuity of 21 41 retract 160 176 Retract, deformation 161 Retract, strong deformation 161 Retraction 160 Retraction, deformation 161 Retraction, strong deformation 161 Reverse path 153 Revolution, surface of 45 Riemann surface 9 Riemannian geometry 8 Riemannian manifold 8 Right action 59 Right action of fundamental group 245 Right coset 354 Right inverse 343 346 Right translation 59 Right-handed 107 Rigid body 13 RING (category of rings) 171 Rings, category of 171 Rings, commutative, category of 171 rotating 134 Russell, Bertrand 338 Russell’s paradox 338 339 Sandwich, ham 254 Sandwich, tofu 254 Saturated 52 Scheme, vertex 96 Schonflies theorem 104 Second category 86 Second countable 32 Second countable, metric space 38 Second countable, product space 50 Second countable, quotient 54 Second countable, sub space 42 Section 184 Section, local 184 236 Section, local, of a covering map 236 Segment, line 347 Seifert — Van Kampen theorem 211 Seifert — Van Kampen theorem and presentations 211 Seifert — Van Kampen theorem, proof 221 Seifert — Van Kampen theorem, special cases 212 217 Semilocally simply connected 265 Separation of a space 65 SEQUENCE 345 Sequence, convergent in a metric space 349 Sequence, convergent in a topological space 20 Sequence, finite 345 Sequence, limit in a discrete space 20 Sequence, limit in a metric space 349 Sequence, limit in a topological space 20 Sequence, limit in a trivial space 20 Sequentially compact 77 Sequentially compact, vs. compact 78 Sequentially compact, vs. limit point compact 77 78 Set 338 SET (category of sets) 171 Set of all sets 339 Set, difference 339 Set, membership 338 Set, theory, naive 337 Sets, category of 171 sgn 353 Sheet 9 237 Short exact sequence 296 Shrinking lemma 82 Side of a geodesic 274 Side of a simplex 108 SIMP (category of simplicial complexes) 171 Simple graph 101 Simple ordering 341 simplex 92 Simplex, abstract 96 Simplex, affine singular 293 Simplex, euclidean 96 Simplex, open 93 Simplex, oriented 106 Simplex, singular 292 Simplex, standard 292 Simplices see “Simplex” Simplicial boundary 324 Simplicial boundary operator 323 Simplicial chain 323 Simplicial complex, abstract 96 Simplicial complex, dimension 334 Simplicial complex, Euclidean 93 Simplicial complex, fundamental group 230 Simplicial complexes, category of 171 Simplicial cycle 324 Simplicial homology groups 324 Simplicial homology groups of a simplex 324 Simplicial homology groups vs. singular 326 Simplicial isomorphism 96 Simplicial map 93 95 Simplicial map between abstract complexes 96 Simplicial Mayer — Vietoris sequence 325 Simply connected 156 Simply connected, covering 261 Simply connected, locally 262 Simply connected, semilocally 265 Sine curve, topologist’s 69 72 88 Singleton 339 Singular boundary 293 294 Singular boundary operator 293 Singular chain 293 Singular chain group 293 Singular cochain 329 Singular cohomology group 329 Singular cycle 294 Singular homology groups 295 Singular homology groups of a contractible space 303 Singular homology groups of a disconnected space 298 Singular homology groups of a one-point space 299 Singular homology groups of spheres 309 Singular homology groups, homotopy invariance 300 Singular homology groups, vs. simplicial 326 Singular homology groups, zero-dimensional 298 Singular map 292 Singular Mayer — Vietoris theorem 292 Singular point 12 Singular point of vector field 192 Singular point, isolated 192 Singular simplex 292 Singular simplex, affine 293 Singular subdivision operator 316 Size, not a topological property 22 Skeleton of a Euclidean simplicial complex 95 Skeleton of an abstract simplicial complex 96 Smale, Stephen 5 7 Small chain 315 Smallest element 341 Smooth manifold 33 SO(n) (special orthogonal group) 11
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