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Lee J.M. — Introduction to Topological Manifolds
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.


Язык: en

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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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 $\mathbb{R}^n$      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
1 2 3 4 5 6
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