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Kinsey L.C. — Topology of surfaces
Kinsey L.C. — Topology of surfaces

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Название: Topology of surfaces

Автор: Kinsey L.C.

Аннотация:

This text was written as an antidote to topology courses such as Spivak describes. It is meant to provide the student with an experience in geometric topology. Traditionally, the only topology an undergraduate might see is point-set topology at a fairly abstract level. The next course the average student would take would be a graduate course in algebraic topology, and such courses are commonly very homological in nature, providing quick access to current research, but not developing any intuition or geometric sense. I have tried in this text to provide the undergraduate with a pragmatic introduction to the field, including a sampling from point-set, geometric, and algebraic topology, and trying not to include anything that the student cannot immediately experience. The exercises are to be considered as an integral part of the text and, ideally, should be addressed when they are met, rather than at the end of a block of material. Many of them are quite easy and are intended to give the student practice working with the definitions and digesting the current topic before proceeding. The appendix provides a brief survey of the group theory needed. The choice of topics addressed is my own, and fairly random at that, and was dictated by the fact that the text is used in a semester course. I often augment the text with articles from the general scientific and undergraduate math journals, which is a gentle way of getting the student to read and discuss mathematics.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 270

Добавлена в каталог: 11.03.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$T_{i}$ space      46
Abelian group      247
Alexander homed sphere      211
Algebraic topology      6 91 117
Antipodal function      226
Antipodal points      32 165
Antipode-preserving function      203
Associated surface      108
Barycentric subdivision      75 172
Base space      162
Basic neighborhood      162
Basis of group      251
Basis of topology      37
Basis, equivalent      40
Betti numbers      142
Borsuk — Ulam theorem      33 203
Boundary of cell      121
Boundary of chain      122
Boundary of manifold      66
Boundary of simplex      126
Boundary of surface      66
Boundary, group      124
Boundary, operator      123
Bounded set      11
Brouwer fixed point theorem      32 199
cell      56
Cell, boundary of      121
Cell, directed      118
Cellular function      146
Cellular function and chain groups      148
Cellular function and homology groups      155
Center      229
Chain      118
Chain function      156
Chain homotopy      179
Chain, boundary of      122
Chain, group      120
Chain, homologous      129
Chain, trivial      130
Classification of surfaces      79
Classification of surfaces with boundary      87
Closed form      245
Closed set      10 39
Closure of a set      11
Cohomology      245
Common subdivision      95 173
Commutative diagram      153
Compact set      44
Compact set, sequentially      26
COMPLEX      57
Complex, dimension of      57
Complex, directed      118
Complex, disjoint union      139
Complex, euler characteristic of      97
Complex, homology groups of      131
Complex, realization of      58
Complex, simplicial      73
Complex, underlying space      58
Complex, underlying space, wedge product      140
Connected set      5 29 43
Connected sum      69
Constant path      194
Constant vector field      227
Continuous function      21 42
Contour lines      223
Contractible space      187
Cover, finite      44
Cover, open      44
Covering space      162
Covering space and euler characteristic      167
Covering space, sheets of      162
Critical point      221
Critical point, index of      228
Critical point, isolated      221
CYCLE      92 124
Cycle, group      124
Deformation retraction      187
Degree      225
Diagram, commutative      153
Diameter, of simplex      177
Diffeomorphism      233
Differential      233 242
Differential form      241
Differential form, closed      245
Differential form, cohomologous      246
Differential form, exact      245
Dimension of complex      57
Dipole      229
Direct sum      250
Directed complex      118
Disc      8
Disc neighborhood      8
Discrete topology      41
Disjoint union      139
Domain      19
Elliptic sector      229
Epimorphism      23
Equivalence, class      52 252
Equivalence, relation      3 251
Equivalence, topological      4 24
Equivalent bases      40
Euler characteristic and 3-manifolds      217
Euler characteristic and betti numbers      142
Euler characteristic of complex      97
Euler characteristic of covering space      167
Euler characteristic of graph      93
Euler characteristic of sphere      98
Euler characteristic, topological invariance of      105
Exact form      245
Exterior point      9
Exterior point, relative      18
Extrinsic topology      63
Face      56
Finite-complement topology      46
Fixed point      32 200
Flow      223
Focus      229
Form, differential      241
Four-color conjecture      109
Frontier of a set      11
Frontier point      9
Function      19
Function, antipodal      226
Function, antipode-preserving      203
Function, cellular      146
Function, chain      156
Function, continuous      21 42
Function, degree      225
Function, identity      23
Function, induced on homology      155 178
Function, inverse      23
Function, one-to-one      23
Function, onto      23
Function, simplicial      151
Function, smooth      233
Fundamental group      195
Fundamental group and homology      197
Fundamental theorem of calculus      244
Generator      248
Genus of surface      107
Genus of surface with boundary      108
Geometric topology      5
Graph      91
Graph, complete      96
Graph, euler characteristic of      93
Green's theorem      244
Group      247
Group, abelian      247
Group, basis for      251
Group, direct sum      250
Group, finitely generated      249
Group, generator      248
Group, presentation      249
Group, rank      251
Group, relation      248
Group, trivial      248
Half-open topology      40
Ham sandwich theorem      206
Handle      68
Hauptvermutung      173
Hausdorff space      46
Heawood conjecture      115
Heawood number      113
Heine — Borel theorem      27 44
Homeomorphism      4 24
Homology      129
Homology and euler characteristic      142
Homology and fundamental group      197
Homology and orientability      136
Homology of disjoint union      139
Homology of Klein bottle      135 137
Homology of sphere      133
Homology of wedge product      140
Homology, groups      131
Homology, induced homomorphism on      155 177
Homology, invariance      172 183
Homology, singular      197
Homomorphism      252
Homotopy      184
Homotopy group      195
Homotopy of paths      190
Homotopy, chain      180
Hyperbolic sector      229
Identification, space      52
Identification, topology      54
Independent      251
INDEX      228
Indiscrete topology      41
Integration      241
Interior of a set      11
Interior, point      9
Interior, point, relative      17
Intrinsic topology      63
Isolated point      9
Isomorphism      252
Jordan curve theorem      208
Kernel      253
Klein bottle      67
Klein bottle, coloring      115
Klein bottle, homology groups      135 137
Klein bottle, planar diagram of      67
Klein bottle, vector field on      239
Limit point      9
Limit point of a sequence      14
Limit point, relative      18
Loop      92 130
Loop, based at x      195
Manifold      64
Manifold with boundary      66
Manifold, 3-dimensional      212
Manifold, non-orientable      68 219
Manifold, smooth      233
Map-coloring problems      109
Matrix      254
Mesh, of simplicial complex      177
Metric      36
Metric, space      36
Metric, topology      37
Minimal triangulation      109
MODULO      252
Moebius band      63
Moebius band, homology of      188
Monomorphism      23
Neighborhood      37
Neighborhood, basic, of covering space      162
Neighborhood, disc      8
Neighborhood, relative      16 41
Non-orientable 3-manifold      219
Non-orientable surface      68
Normal space      46
One-point union      140
One-to-one      23
Onto      23
Open cover      44
Open set      10 37
Open set, relative      18
Orientation and homology      136
Orientation of cell      118
Orientation of simplex      126
Pancake theorem      33
Parabolic sector      229
Path      190
Path, constant      194
Path, homotopy of      190
Path, product of      192
Permutation      168
Phase portrait      222
Planar diagram      62
Platonic solid      101
Poincare conjecture      198
Poincare Index Theorem      230 236
Poincare — Bendixson theorem      229
Point-set topology      5 7
Polygonal curve      208
Polyhedron, regular      101
Presentation, group      249
Product of paths      192
Product space      47
Projective plane      71
Projective plane, coloring      113
Projective plane, cover of      165 203
Projective plane, planar diagram of      71
Projective plane, vector field on      237
Pseudomanifold      217
Quotient space      51
Radial projection      111 188
RANGE      19 253
Rank      251
Realization      58
Regular polyhedron      101
Regular space      46
Relation      248
Relative, exterior point      18
Relative, interior point      17
Relative, limit point      18
Relative, neighborhood      16 41
Relative, open set      18
Retraction      186 199
Retraction, deformation      187
Saddle point      229
Schoenflies theorem      211
Separation axioms      46
SEQUENCE      14
Set-theoretic inverse      21
Sheets of covering space      162
simplex      73 126
Simplex, boundary of      126
Simplex, diameter      177
Simplex, oriented      126
Simplicial approximation      176
Simplicial approximation theorem      178
Simplicial complex      73
Simplicial complex, mesh      177
Simplicial complex, vector field on      240
Simplicial function      151
Singular homology      197
Sink      229
Skeleton      58
Smooth, function      233
Smooth, manifold      233
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