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Lee J.M. — Introduction to Topological Manifolds
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Название: Introduction to Topological Manifolds
Автор: 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.
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Рубрика: Математика /Геометрия и топология /Общая топология /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Год издания: 2000
Количество страниц: 385
Добавлена в каталог: 19.04.2005
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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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