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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
First category      86
First countable      32
First countable, locally Euclidean spaces      38
First countable, metric spaces      38
First isomorphism theorem      356
Five lemma      313
Fixed point theorem, Brouwer      192
Folding      134
Forgetful functor      172
Formal linear combination      97
Free Abelian group      204
Free abelian group, characteristic property      204
Free abelian group, uniqueness      208
Free abelian subgroup      205
Free group      200
Free group action      60
Free group, characteristic property      200
Free group, uniqueness      201
Free product      195
Free product, characteristic property      198
Free product, uniqueness      199
Free vector space      97
Freedman, Michael      7
Freedom, degrees of      2
Full subcategory      171
Function      341
Function, multiple-valued      8
Functor, cohomology      330
Functor, contravariant      172
Functor, covariant      172
Functor, exact      335
Functor, forgetful      172
Functor, fundamental group      172
Functor, homology      296
Fundamental group      6 152
Fundamental group and homology      305
Fundamental group and surface presentation      217
Fundamental group is a group      154
Fundamental group of a graph      215
Fundamental group of a manifold is countable      189
Fundamental group of a polyhedron      230
Fundamental group of a product      189
Fundamental group of a surface      220
Fundamental group of a surface, abelianized      228
Fundamental group of a topological group      191
Fundamental group of a wedge of spaces      213
Fundamental group of spheres      188 217
Fundamental group of the circle      180
Fundamental group of the projective plane      247
Fundamental group of the torus      189
Fundamental group, action on fiber      245
Fundamental group, change of base point      154
Fundamental group, functor      172
Fundamental group, homotopy invariance      161
Fundamental group, topological invariance      159
Fundamental theorem of algebra      191
Gauss — Bonnet theorem      8
General linear group      10 59 60
General linear group, complex      10
General position      92
General relativity      14
Generating circle      45
Generator of a cyclic group      356
Generator of a group      199
Generator of a presentation      201
Genus      144
Geodesic, hyperbolic      271
Geodesic, polygon      273
Geodesic, polygon, regular      274
Geometric realization      97 99
Geometric realization of a polygonal presentation      131
Geometric realization, functor      172
Geometrization conjecture      7
Geometry, algebraic      11
Geometry, Euclidean      7
Geometry, plane      7
Geometry, Riemannian      8
Geometry, solid      7
Gluing lemma      46
Gluing lemma, counterexample      62
Graph      100
Graph of a continuous function      2 43
Graph of a continuous functionis a manifold      44
Graph of a relation      8
Graph, finite      101
Graph, fundamental group      215
Graph, simple      101
Gravitation      14
Gravity, center of      110
Greatest lower bound      343
Group      352
GROUP (category of groups)      171
Group as a category      173
Group, abelian      11 203
Group, action      59 266
Group, action, continuous      60
Group, action, free      60
Group, action, proper      266
Group, action, quotient by      61
Group, action, transitive      60
Group, automorphism, of covering      248
Group, complex general linear      10
Group, complex special linear      11
Group, covering      248
Group, direct product      353
Group, discrete      58
Group, divisible      335
Group, free      200
Group, free abelian      204
Group, fundamental      6 152
Group, general linear      10 59 60
Group, homotopy      169—170
Group, injective      335
Group, Lie      10
Group, orthogonal      11 59 60
Group, permutation      353
Group, presentation      201 202
Group, special linear      11
Group, special orthogonal      11
Group, special unitary      11
Group, symmetric      353
Group, theory, combinatorial      203
Group, topological      58
Group, topological, quotient      63
Group, unitary      11
Groups, category of      171
Hairy ball theorem      322
Half space, upper      34
Ham sandwich theorem      254
handedness      106
Handle      9 129
Hauptvermutung      112
Hausdorff, if diagonal is closed      62
Hausdorff, product space      50
Hausdorff, space      31
Hausdorff, sub space      42
Heegaard, Poul      137
Heine — Borel theorem      351
hole      5 6 147
Holomorphic      8
Hom(X, Y) (set of group homomorphisms)      173
Homeomorphic      4 22
Homeomorphic is an equivalence relation      22
Homeomorphism      4 22
Homeomorphism, local      24
Homeomorphism, local, openness      24
Homeomorphism, vs. closed map      27
Homeomorphism, vs. open map      27
Homogeneity of norm      89
Homogeneous, space      63
Homogeneous, spacetime      14
Homological algebra      297
Homological degree      320
Homologous      295
Homology and the fundamental group      305
Homology of a compact polyhedron      334
Homology of a contractible space      303
Homology of a disconnected space      298
Homology of a one-point space      299
Homology of a pseudomanifold      334
Homology of a simplex      324
Homology of a triangulable manifold      334
Homology of a wedge      334
Homology of spheres      309
Homology, class      295
Homology, functor      296
Homology, groups of a chain complex      297
Homology, groups, simplicial      324
Homology, groups, singular      295
Homology, homomorphism, induced by a chain map      297
Homology, homomorphism, induced by a continuous map      296
Homology, homotopy invariance      300
Homology, sequence, long exact      311
Homology, simplicial      324
Homology, simplicial, vs. singular      326
Homology, singular      295
Homology, singular, vs. simplicial      326
Homology, topological invariance      296
Homology, zero-dimensional      298
Homomorphism from a quotient group      355
Homomorphism of cyclic group      357
Homomorphism of group      354
Homomorphism of topological groups      270
Homomorphism, covering      258
Homomorphism, covering is a covering map      258
Homomorphism, criterion, covering      260
Homomorphism, fundamental group, induced by a continuous map      159
Homomorphism, homology, induced by a chain map      297
Homomorphism, homology, induced by a continuous map      296
Homotopic      148
Homotopic, degree      320
Homotopic, maps and fundamental group homomorphisms      164
Homotopic, maps and homology homomorphisms      300
Homotopic, path      151
Homotopic, relative to a subspace      151
Homotopy is an equivalence relation      148
Homotopy of maps      148
Homotopy, category      173
Homotopy, category, pointed      173
Homotopy, chain      303 317
Homotopy, equivalence      161
Homotopy, equivalence and deformation retraction      166
Homotopy, equivalence is an equivalence relation      161
Homotopy, equivalent      161
Homotopy, groups      169—170
Homotopy, invariance of path product      152
Homotopy, invariance of singular homology      300
Homotopy, invariance of the fundamental group      161
Homotopy, lifting property      238
Homotopy, lifting property of the circle      181
Homotopy, path      151
Homotopy, path and composition      158
Homotopy, path is an equivalence relation      151
Homotopy, preserved by composition      149
Homotopy, relative      151
Homotopy, straight-line      150
Homotopy, theory      170
Homotopy, type      161
Hull, convex      92
Hurewicz, homomorphism      308
Hurewicz, theorem      308
Hurewicz, Witold      308
Hyperbolic disk      271
Hyperbolic geodesic      271
Hyperbolic metric      271
Hyperbolic metric, triangle inequality      289
Hyperbolic neighborhood, regular      278
Hyperplane, affine      92
i (imaginary unit)      344
I (unit interval)      54
Ideal point      12
Identification space      52
Identity in a category      171
Identity in a group      352
Identity in a group, uniqueness      353
Identity, map      342
Identity, map, continuity      21
Identity, path class      153
Image is a subgroup      354
Image of a function      342
Image of a homomorphism      354
Image of a normal subgroup      356
Image, inverse      342
Image, set      342
Imaginary unit      344
Imf (image of f)      354
Inclusion map      342
Inclusion map, continuity      41
Increasing function      345
Independent, linearly      204
Index of a subgroup      354
Index of a vector field      192
Index set      345
Indexed collection      345
Indexed collection, intersection      345
Indexed collection, union      345
Induced homomorphism of fundamental groups      159
Induced homomorphism of fundamental groups by homotopic maps      164
Induced homomorphism, in homology      296 297
Induced morphism      172
Induced orientation      107
Induced subgroup      239
Infimum      343
Infinite cyclic group      200 356
Infinite dimensional simplicial complex      96
Infinite product      49
Infinite set      344
Initial point of a path      150
Initial vertex      131
Injection in a category      175
Injection into direct sum      177
Injection into disjoint union      345
Injection into free group      200
Injection into free product      197
Injective      342
Injective, group      335
Injectivity theorem      239
Inside out sphere      5
int      see “Interior”
Integers      344
Integers modulo n      356
Interior      25 26
Interior of a manifold with boundary      34 38
Interior of a simplex      93
Intermediate Value Theorem      65 68
Intersection of an indexed collection      345
Intersection of closed sets in a metric space      349
Intersection of closed sets in a topological space      24
Intersection of open sets in a metric space      349
Intersection of open sets in a topological space      18
Intersection of sets      339
Intertwined edge pairs      140
interval      344
Interval is connected      68
Interval, unit      54
Invariance of dimension      318 319
Invariant, combinatorial      113 142
Invariant, topological      6
Inverse in a group      352
Inverse in a group, uniqueness      353
1 2 3 4 5 6
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