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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
Component      70 347
Component is closed      71
Component of ordered pair      340
Component, functions      346
Component, path      72
composition      342
Composition in a category      170
Composition of quotient maps      53
Composition, continuity of      20 21
Cone      110
Cone on an affine simplex      315
conformal      273
Congruence modulo a subgroup      354
Conjugacy class      354
Conjugacy theorem      243
Conjugate subgroups      354
Conjugation      354
Connected edge path      101
Connected interval      68
Connected locally      72
Connected locally path      72
Connected product space      67
Connected quotient space      67
Connected simply      156
Connected space      65
Connected subset      66
Connected subset of $\mathbb{R}$      68
Connected sum      126
Connected sum is a manifold      126
Connected sum with sphere      129
Connected sum, covering of      253
Connected sum, polygonal presentation      136
Connecting homomorphism      309 310
Connecting homomorphism, naturality      312
Connectivity relation      70
Connectivity relation, path      71
Consistent orientations      107
Consolidating      134
Constant loop      151
Constant map, continuity of      21
Continuity and closed sets      25
Continuity and convergent sequences      38 349
Continuity at a point      21
Continuity between Euclidean spaces      347
Continuity between metric spaces      349
Continuity between topological spaces      20
Continuity in terms of basis      29
Continuity of composition      20 21
Continuity of constant map      21
Continuity of identity map      21
Continuity of restriction      21
Continuity, local criterion      21
Continuity, open set criterion      350
continuous      see “Continuity”
Continuous deformation      4
Continuous group action      60
Continuous image of compact set      73
Continuous image of connected set      67
Continuous map induced by a simplicial map      99
Contractible space      161
Contractible space is simply connected      162
Contractible space, singular homology of      303
Contravariant functor      172
Convergent sequence in a metric space      349
Convergent sequence in a topological space      20
Convergent sequence in Euclidean space      347
Convergent sequence is Cauchy      350
Convergent sequence vs. continuity      38 349
Convex      69
Convex, hull      92
Convex, set      176
Convex, set, homeomorphic to ball      80
Convex, set, simply connected      156
Coordinate      347
Coordinate chart      31
Corners      23
Correspondence, one-to-one      342
Coset, left      354
Coset, multiplication of      355
Coset, right      354
Coset, space      61
Countable, basis      32
Countable, dense subset      38
Countable, dense subset of $\mathbb{R}^n$      27
Countable, first      32
Countable, neighborhood basis      32
Countable, second      32
Countable, set      344
Countable, subcover      32
Countable, subset      344
Countable, union      344
Countably infinite      344
Counterclockwise      107
Covariant functor      172
Cover      32
Cover, open      32 73
Cover, open, in a metric space      350
Cover, open, of a subset      73
Covering of connected sum      253
Covering of Klein bottle      253
Covering of lens space      286
Covering of manifold      253
Covering of projective space      235 253
Covering of torus      270 286
Covering, cardinality of fibers      236 247
Covering, classification      283
Covering, group      248
Covering, group, structure theorem      250
Covering, group, transitivity      249
Covering, homomorphism      258
Covering, homomorphism, criterion      260
Covering, homomorphism, is a covering map      258
Covering, isomorphism      258
Covering, isomorphism theorem      260
Covering, map      234
Covering, map, classification      283
Covering, map, is a local homeomorphism      235
Covering, map, is a quotient map      235
Covering, map, is open      235
Covering, map, product of      253
Covering, space      234
Covering, space, universal      261
Covering, transformation      248
Covering, uniqueness      260
Covering, universal      261
Crease      5
CRING (category of commutative rings)      171
Crunch, big      14
cube      4
Cube, closed      351
Cube, closed, nested      351
Cube, open      29
Cubical surface      23
Cup, coffee      5
Curvature      7
Curve      2 117
Curve, classification      118
Curve, plane      2
Curve, space      2
Curve, space-filling      188
Cusp      11
Cutting      134
Cycle in a graph      163
Cycle, simplicial      324
Cycle, singular      294
Cyclic group      356
Cyclic group, homomorphism      357
Cyclic group, infinite      200
Cyclic subgroup      356 357
Cylinder      3
Cylinder, mapping      167
Deck transformation      248
Deformation      148
Deformation and homotopy equivalence      166
Deformation, continuous      4
Deformation, retract      161
Deformation, retract, strong      161
Deformation, retraction      161
Deformation, strong      161
Degree of a map      191 320
Degree of a map, homological      320
Degree of a map, homotopic      320
Degrees of freedom      2
Dehn, Max      137 203
Dense      26
Dense, nowhere      85
Descending to the quotient      56 57
Descending to the quotient, homomorphism      356
Diagonal      62
Diagram, commutative      355
Diameter      76 349
Difference of sets      339
DIMENSION      1
Dimension of a Euclidean simplicial complex      94
Dimension of a manifold      33
Dimension of a simplex      92
Dimension of a simplicial complex      334
Dimension of an abstract simplex      96
Dimension of an abstract simplicial complex      96
Dimension of an affine subspace      92
Direct product      353
Direct sum      177
Disconnected      65
Discontinuous, properly      268
Discrete group      58
Discrete metric      348
Discrete space      19
Discrete space, closed sets      25
Discrete subgroup      270
Discrete topology      19
Disjoint sets      339
Disjoint union      340 345
Disjoint union, topology      37 177
Disjoint union, topology, characteristic property      62
Disk, Euclidean      31
Disk, hyperbolic      271
Distance function      348
Divisible group      335
Domain      341
Dot product      347
Doughnut surface      3 5 45
Doughnut surface, homeomorphic to torus      51 80
Dual map      173
Dual space      173
Dynamical system      13
Edge of a presentation      131
Edge of a simplex      93
Edge, pairing transformation      274
Edge, path      101
Edge, path, connected      101
Edge, path, periodic      119
Edge, path, reduced      101
Edge, point      124
Einstein, Albert      14
Einstein, Albert, field equations      14
Einstein, Albert, general relativity      14
Element of a set      338
Elementary particle      14
Elementary reduction      195
Elementary subdivision      112
Elementary transformation      133
Ellipsoid      3
Embedding      40
Empty set      338
Empty set, existence      343
Empty word      194
Equilibrium point      14
Equivalence of words      195
Equivalence, class      52 340
Equivalence, combinatorial      112
Equivalence, homotopy      161
Equivalence, homotopy is an equivalence relation      161
Equivalence, relation      52 340
Equivalence, relation, generated by a relation      340
Equivalence, topological      4 22
Euclidean ball      31
Euclidean ball, regular      83
Euclidean disk      31
Euclidean dot product      347
Euclidean geometry      7
Euclidean locally      4 30
Euclidean metric      348
Euclidean neighborhood      30
Euclidean polyhedron      94
Euclidean simplex      96
Euclidean simplicial complex      93
Euclidean space      2 347
Euclidean space is second countable      33
Euclidean space, ambient      17
Euclidean space, zero-dimensional      31 347
Euclidean topology      19
Euclidean triangle      7
Euler characteristic      113 142 328
Euler characteristic and cohomology      333
Euler characteristic of a graph      230
Euler characteristic of a topological space      328
Euler characteristic of compact surfaces      143 229
Euler characteristic, combinatorial invariance      113 142
Euler characteristic, topological invariance      229 327
Euler’s formula      112
Even map      253
Even permutation      353
Evenly covered      184 234
Exact functor      335
Exact homology sequence      311
Exact sequence      296
Exact sequence of chain complexes      310
Exact sequence, long      311
Exact sequence, short      296
Existence of real numbers      343
Existence, axiom      343
Exponential function      20
Exponential quotient map      61 179 235
Ext      see “Exterior”
Extension Lemma      331
Extension of a map      342
Exterior      25 26
Extreme Value Theorem      73 76 352
Face of a presentation      131
Face of a simplex      93
Face of an abstract simplex      96
Face, boundary      93
Face, map      293
Face, point      124
Face, proper      93
Family      338
Fan transformation      125
Fiber      53
Fiber, action on, by fundamental group      245
Field      330
Field equations, Einstein      14
Field, characteristic zero      330
Field, ordered      343
Figure eight space      55 160
Finite graph      101
Finite locally      96
Finite sequence      345
Finite set      344
Finite simplicial complex      96
Finitely presented      202
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