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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
Inverse of a path class      153
Inverse, image      342
Inverse, left      343
Inverse, map      342
Inverse, right      343 346
Isolated singular point      192
Isometry      8
Isomorphic coverings      258
Isomorphism in a category      171
Isomorphism of coverings      258
Isomorphism of groups      354
Isomorphism, problem      203
Isomorphism, simplicial      96
Isomorphism, theorem, covering      260
Isomorphism, theorem, first      356
Isotropic spacetime      14
k-skeleton      see “Skeleton”
Ker      see “Kernel”
Kernel      354
Kernel is a subgroup      354
Kernel is normal      355
Klein bottle      126
Klein bottle, covering      253
Klein bottle, presentation      133
largest element      341
Latitude      3
Laws of motion, Newton’s      12
Least upper bound      343
Lebesgue number      76
Lebesgue number, lemma      76
Left action      59
Left coset      354
Left coset space      61
Left inverse      343
Left translation      59 63
Length      7 347
Lens space      269
Lens space, coverings of      286
Lie group      10
Lie group, abelian      11
Lift      179 180 237
Lifting criterion      240
Lifting problem      239
Lifting property, homotopy      238
Lifting property, homotopy of the circle      181
Lifting property, path      238
Lifting property, path of the circle      181
Lifting property, unique      237
Lifting property, unique of the circle      181
Limit of a sequence in a discrete space      20
Limit of a sequence in a Hausdorff space      32
Limit of a sequence in a metric space      349
Limit of a sequence in a topological space      20
Limit of a sequence in a trivial space      20
Limit point      26 348
Limit point and closed sets      26
Limit point, compact      76
Limit point, compact vs. compact      77 78
Limit point, compact vs. sequentially compact      77 78
Line with two origins      62
Line, long      88
Line, segment      347
linear combination      204
Linear combination, formal      97
Linear ordering      341
Linear transformation      20
Linearly independent      204
Local criterion for continuity      21
Local finiteness      see “Locally finite”
Local homeomorphism      24
Local homeomorphism, openness      24
Local section      184 236
Local section of a covering map      236
Locally compact      81
Locally compact, Hausdorff space      81 82 89
Locally connected      72
Locally Euclidean      4 30
Locally Euclidean, implies first countable      38
Locally finite      93 96
Locally path connected      72
Locally simply connected      262
Logarithmic function      20
Long exact homology sequence      311
Long line      88
longitude      3
Loop      151
Loop, based at a point      151
Loop, constant      151
Lorentz metric      14
Lower bound      341
Main theorem on compactness      73
Main theorem on connectedness      67
Manifold      1 4 33
Manifold is locally compact Hausdorff      83
Manifold is locally path connected      72
Manifold with boundary      34 334
Manifold with boundary, 2-dimensional      191
Manifold, boundary      35
Manifold, classification      6
Manifold, complex      33
Manifold, countable fundamental group      189
Manifold, homology of      334
Manifold, product of      51
Manifold, Riemannian      8
Manifold, smooth      33
Manifold, topological      33
Manifold, zero-dimensional      37
MAP      341
Mapping      341
Mapping cylinder      167
Markov, A. A.      7
Mathematical object      338
Maximal      341
Maximal tree      214
Mayer — Vietoris sequence, cohomology      335
Mayer — Vietoris sequence, simplicial      325
Mayer — Vietoris sequence, singular      309
Mayer — Vietoris theorem, cohomology      335
Mayer — Vietoris theorem, simplicial      325
Mayer — Vietoris theorem, singular      292 309
Mechanics, classical      12
Member of a set      338
membership      338
mesh      317
Metric      348
Metric, discrete      348
Metric, Euclidean      348
Metric, hyperbolic      271
Metric, Lorentz      14
Metric, space      348
Metric, space, first countable      38
Metric, space, Hausdorff      31
Metric, space, second countable      38
Metric, space, subspace of      40
Metric, topology      19
Minimal      341
Mobius, band      105 176
Mobius, group      272
Mobius, transformation      272
Modulo n      356
Moise, Edwin      105
Monodromy theorem      239
Morphism      170
Morphism, induced      172
Motion, Newton’s laws of      12
Multiple-valued function      8
Multiplication of cosets      355
Multiplication of path classes      153
Multiplication of paths      152
Multiplication of words      194
Multiplication, group      352
n-dimensional manifold      33
n-dimensional topological manifold      33
n-holed torus      129
n-holed torus, universal covering      275
n-manifold      33
n-sphere      44
n-sphere, singular homology      309
n-torus      51
n-torus as a coset space of $\mathbb{R}^n$      61
n-torus, fundamental group      189
n-tuple, ordered      345
Naive set theory      337
Natural numbers      344
Natural orientation      106
Naturality of connecting homomorphisms      312
Nearness      18
Neighborhood      18
Neighborhood, basis      32
Neighborhood, basis, countable      32
Neighborhood, basis, nested      77
Neighborhood, Euclidean      30
Neighborhood, regular hyperbolic      278
Nested cubes      351
Nested neighborhood basis      77
Nested sets      76
Newton’s laws of motion      12
Nondegenerate base point      212
Nonorientable surface      144
Nonorientable surface, covering of      253
Norm      89 347
Normal closure      201
Normal covering      245
Normal subgroup      354
Normal subgroup, image      356
North pole      3 45
Nowhere dense      85
nth homotopy group      170
nth power map      191 235
Null homotopic      151
O(n) (orthogonal group)      11 59 60
Object in a category      170
Object, mathematical      338
Odd map      253
Odd permutation      353
One-point compactihcation      89
One-point space, singular homology      299
One-point union      55
One-to-one correspondence      342
One-to-one function      342
Onto      342
Open ball      348
Open ball is an open set      349
Open cover      32 73
Open cover in a metric space      350
Open cover of a subset      73
Open map      24
Open map, product of      62
Open map, vs. homeomorphism      27
Open set as a topological space      19
Open set in a metric space      348
Open set in a topological space      18
Open set is a manifold      34
Open set is Hausdorff      31
Open set is second countable      33
Open set, criterion for continuity      350
Open set, intersection, in a metric space      349
Open set, intersection, in a topological space      18
Open set, union, in a metric space      349
Open set, union, in a topological space      18
Open simplex      93
Open star      114
Orbit      60 245
Orbit, criterion      249
Orbit, space      61 266
Orbit, space by free proper group action      268
Order of a group      353
Order topology      37
Ordered field      343
Ordered n-tuple      345
Ordered pair      340
Ordered set, partially      341
Ordered set, totally      37 341
Ordered well      88 341
Ordering, linear      341
Ordering, partial      341
Ordering, simple      341
Ordering, total      341
Orientability is combinatorially invariant      115
Orientable pseudomanifold      334
Orientable simplicial complex      107
Orientable surface      144 229
Orientation of a simplex      105
Orientation of a simplicial complex      107
Orientation, induced      107
Orientation, natural      106
Oriented presentation      144
Oriented simplex      106
Origins, line with two      62
Orthogonal group      11 59 60
Orthogonal group, special      11
Pair, ordered      340
Pancakes      234
parabola      2
Paraboloid      3
Parameters      1
Partial ordering      341
Partially ordered set      341
Particle, elementary      14
Partition      52 340
Passing to the quotient      56 57
Passing to the quotient, homomorphism      356
Pasting      134
Path      69 150
Path, class      151
Path, class identity      153
Path, class inverse      153
Path, class multiplication      153
Path, class multiplication, associativity      153
Path, class product      153
Path, class product, associativity      153
Path, component      72
Path, connected      69
Path, connected, locally      72
Path, connectivity relation      71
Path, homotopic      151
Path, homotopy      151
Path, homotopy and composition      158
Path, homotopy is an equivalence relation      151
Path, lifting property      238
Path, lifting property of the circle      181
Path, multiplication      152
Path, multiplication, grouping      154
Path, multiplication, homotopy invariance      152
Path, product      152
Path, product, grouping      154
Path, product, homotopy invariance      152
Path, reverse      153
Periodic edge path      119
Periodic trajectory      14
Permutation      342
Permutation, even      353
Permutation, group      353
Permutation, odd      353
Plane      3
Plane, curve      2
Plane, geometry      7
Plane, projective      119
Poincare, conjecture      6
Poincare, Henri      4 6
1 2 3 4 5 6
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