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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
$GL(n,\mathbb{R})$ (general linear group)      10 59 60
$Hom_C(X,Y)$ (set of morphisms)      170
$SL(n,\mathbb{C})$ (complex special linear group)      11
$SL(n,\mathbb{R})$ (special linear group)      11
$TOP_{\ast}$ (pointed topological category)      171
$\bigcap_{\alpha}X_{\alpha}$, (intersection)      345
$\bigcup_{\alpha}X_{\alpha}$ (union)      345
$\coprod_{\alpha}X_{\alpha}$ (disjoint union)      345
$\Delta$      see “Boundary”
$\iota_S$ (inclusion map)      342
$\mathbb{B}^n$ (unit ball in $\mathbb{R}^n$)      22
$\mathbb{C}$ (set of complex numbers)      344
$\mathbb{C}\mathbb{P}^n$ (complex projective space)      62
$\mathbb{H}^n$ (upper half space)      34
$\mathbb{N}$ (set of natural numbers)      344
$\mathbb{P}^2$ (projective plane)      119
$\mathbb{P}^n$ (real projective space)      55
$\mathbb{Q}$ (set of rational numbers)      344
$\mathbb{R}$ (set of real numbers)      343
$\mathbb{R}\langleS\rangle$ (free vector space)      97
$\mathbb{R}^n$ (n-dimensional Euclidean space)      347
$\mathbb{S}^n$ (unit n-sphere)      44
$\mathbb{T}^2$ (torus)      51
$\mathbb{T}^n$ (n-torus)      51
$\mathbb{Z}$ (set of integers)      344
$\mathbb{Z}/\langle n\rangle$ (integers modulo n)      356
$\mathbb{Z}/\langle S\rangle$ (free abelian group)      204
$\mathcal{P}(S)$ (power set)      339
$\mathcal{U}$-small chain      315
$\pi_1(X)$ (fundamental group)      155
$\pi_1(X,q)$ (fundamental group)      152
$\varepsilon$ (exponential quotient map)      61 179 235
AB (category of abelian groups)      171
Abelian group      203 353
Abelian group, free      204
Abelian groups, category of      171
Abelian Lie group      11
Abelianization      227
Abelianization of fundamental groups of surfaces      228
Abelianization, characteristic property      227
Abelianization, functor      231
Abelianization, uniqueness      231
Absolute value      20
Abstract simplex      96
Abstract simplicial complex      96
Accumulation point      26 348
Action of a group      59 266
Action of a group, quotient by      61
Action of fundamental group on fiber      245
Action, continuous      60
Action, free      60
Action, left      59 266
Action, proper      266
Action, right      59
Action, transitive      60
Affine chain      293
Affine hyperplane      92
Affine map      92
Affine singular simplex      293
Affine subspace      92
Algebra, fundamental theorem of      191
Algebraic geometry      11
Algebraic topology      6
Algebraic variety      12
Algebraically closed      11
Algorithm, reduction      196
Ambient Euclidean space      17
Analysis situs      4
angle      7 347
Angle, function, for a path in $\mathbb{S}^1$      179 182
Angle-sum theorem      7
Antipodal map      248 321
Antipodal map, degree of      321
Antipodal map, homotopic to identity      322
Antisymmetric relation      341
Area      7
Associative      352
Associativity of path class product      153
Automorphism group of covering      248
Automorphism of covering      248
Automorphism of group      354
Axiom of Choice      346
Axiom, Cartesian product      340
Axiom, existence      343
Axiom, power set      339
Axiom, specification      338
Axiom, union      339
Axioms for set theory      338—347
Baire category theorem      85
Ball, closed      348
Ball, closed is a closed set      349
Ball, closed is a manifold with boundary      62
Ball, Euclidean      31
Ball, Euclidean, regular      83
Ball, open      348
Ball, open is an open set      349
Ball, unit, in $\mathbb{R}^n$      22
Barycenter      110
Barycentric subdivision      110 315
Base of covering      234
Base point      151
Base point, change of      154
Base point, nondegenerate      212
Based at a point      151
Basis and continuity      29
Basis for a topology      29
Basis for discrete topology      29
Basis for Euclidean topology      29
Basis for free abelian group      204
Basis for metric topology      29
Basis for product topology      48 50
Basis for subspace topology      42
Basis for trivial topology      29
Basis in a set      27
Basis of Euclidean balls      38
Basis, countable      32
Basis, criterion      27
Basis, neighborhood      32
Basis, neighborhood, countable      32
Basis, standard, for $\mathbb{Z}^n$      204
Basis, topology generated by      27
Belongs to a set      338
Betti number      328
Big crunch      14
Bijective      342
Body, rigid      13
Bound, lower      341
Bound, upper      341
Boundary      25 26
Boundary of a boundary      294
Boundary of a boundary, simplicial      323
Boundary, face      93
Boundary, manifold with      34
Boundary, of a manifold with boundary      34 38
Boundary, of a simplex      93
Boundary, of a singular simplex      293
Boundary, operator      297
Boundary, operator, simplicial      323
Boundary, operator, singular      293
Boundary, simplicial      324
Boundary, singular      294
Boundary, topological      35
Bounded, above      341
Bounded, below      341
Bounded, not a topological property      22
Bounded, set      349
Bouquet of circles      55
Branch      9
Brouwer fixed point theorem      192 334
Calabi — Yau manifold      15
Cardinality      344
Cardinality of fibers of covering      236 247
Cartesian coordinates      3
Cartesian product      340 346
Cartesian product, axiom      340
Cartesian product, finite      346
Cartesian product, infinite      346
Category      170
Category of Abelian groups      171
Category of commutative rings      171
Category of complex vector spaces      171
Category of groups      171
Category of real vector spaces      171
Category of rings      171
Category of sets      171
Category of simplicial complexes      171
Category of topological spaces      171
Category, first      86
Category, homotopy      173
Category, pointed homotopy      173
Category, second      86
Category, theorem, Baire      85
Cauchy sequence      350
Cauchy sequence vs. convergent sequence      350
Center of a group      208
Center of gravity      110
Chain, affine      293
Chain, complex      297
Chain, complex, homology groups of      297
Chain, group, simplicial      323
Chain, group, singular      293
Chain, homotopic      303
Chain, homotopy      303 317
Chain, map      297
Chain, simplicial      323
Chain, singular      293
Change of base point      154
Characteristic property, abelianization      227
Characteristic property, disjoint union topology      62
Characteristic property, free abelian group      204
Characteristic property, free group      200
Characteristic property, free product      198
Characteristic property, product topology      49
Characteristic property, quotient topology      56
Characteristic property, subspace topology      41
Characteristic zero      330
Characterization of quotient, maps      57
Chart      31
Chart on a manifold with boundary      34
Chart, coordinate      31
Choice function      346
Choice, axiom of      346
circle      2 45
Circle as coset space of $\mathbb{R}$      61
Circle, fundamental group      180
Circle, generating      45
Circle, homotopy lifting property      181
Circle, path lifting property      181
Circle, representative      157
Circle, unique lifting property      181
Circle, unit      45
class      339
Class, equivalence      340
Classical mechanics      12
Classification of 1-manifolds      118
Classification of 2-manifolds      6 137 229
Classification of 3-manifolds      7
Classification of coverings      283
Classification of manifolds      6
Classification of n-manifolds      7
Classification of surface presentations      137
Classification of surfaces      6 137 229
Classification of torus coverings      286
Closed ball      348
Closed ball is a closed set      349
Closed ball is a manifold with boundary      62
Closed cube      351
Closed map      27
Closed map, lemma      79
Closed map, product of      62
Closed map, vs. homeomorphism      27
Closed set      24 26
Closed set and continuity      25
Closed set and limit points      26
Closed set in a compact space      74
Closed set in a discrete space      25
Closed set in a metric space      349
Closed set in a subspace      41
Closed set, intersection, in a metric space      349
Closed set, intersection, in a topological space      24
Closed set, union, in a metric space      349
Closed set, union, in a topological space      24
Closure      25 26
Closure and sequences      38
Closure, normal      201
Cluster point      26 348
Coboundary      329
Coboundary, operator      329
Cochain, complex      329
Cochain, map      330
Cochain, singular      329
Cocycle      329
Codomain      342
Coffee cup      5
Coherent topology      99
Coherent topology of finite union      114
Cohomology with field coefficients      331 332
Cohomology, functor      330
Cohomology, Mayer — Vietoris sequence      335
Cohomology, singular      329
Cohomology, topological invariance      330
Collection      338
Combinatorial equivalence      112
Combinatorial group theory      203
Combinatorial invariant      113 142
Combinatorial invariant, Euler characteristic      113 142
Combinatorial invariant, orientability      115
Commutative diagram      355
Commutative rings, category of      171
Commutator subgroup      227
Compact vs. limit point compact      77 78
Compact vs. sequentially compact      78
Compact, implies closed and bounded      350
Compact, limit point      76
Compact, locally      81
Compact, relatively      82
Compact, sequentially      77
Compact, set, continuous image      73 352
Compact, set, continuous image in a Hausdorff space      74
Compact, set, continuous image in a metric space      350
Compact, topological space      73
Compact, topological space, product of      74
Compact, topological space, quotient of      74
Compactification, one-point      89
Complement      339
Complementary edge pair      139
Complete, metric space      350
Complete, ordered field      343
Complex, analysis      8
Complex, analytic      8
Complex, chain      297
Complex, general linear group      10
Complex, manifold      33
Complex, numbers      344
Complex, projective space      12 62
Complex, simplicial, abstract      96
Complex, simplicial, category of      171
Complex, simplicial, Euclidean      93
Complex, special linear group      11
Complex, vector spaces, category of      171
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