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Lima E.L. — Fundamental Groups and Covering Spaces
Lima E.L. — Fundamental Groups and Covering Spaces

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Название: Fundamental Groups and Covering Spaces

Автор: Lima E.L.

Аннотация:

The elementary character of fundamental groups and covering spaces are presented as suitable for introducing algebraic topology. The two topics are treated in separate sections. The focus is on the use of algebraic invariants in topological problems. Applications to other areas of mathematics such as real analysis, complex variables, and differential geometry are also discussed.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 214

Добавлена в каталог: 11.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Action of a group      154
Action, orbit of an      154
Action, transitive      154
Analytic continuation      124
Angle element      109
Base point      20 33
Change of variable formula      110
Circle eversion      104
Coherent atlas      186
Complex projective space      67
Complex projective space as a locally trivial fibration      69
Complex projective space, fundamental group of      71
Convenient neighborhood      130
Covering      125
Covering, automorphism      162
Covering, base of a      125
Covering, distinguished neighborhood      125
Covering, endomorphism      162
Covering, fiber of a      125
Covering, homomorphic      160
Covering, homomorphism      161
Covering, isomorphism      162
Covering, number of leaves of a      126
Covering, oriented double      192
Covering, product      193
Covering, regular      153
Covering, transformation      162
Covering, translation      162
Covering, universal      67 165
Curve, eversion of a      103 104
Curve, normal vector field to a      103 104
Curve, orientation of a      103
Curve, Peano      42
Curve, rotation number of a      99 103
Curve, simple      103
Curve, Space filling      42
Curve, winding number of a      95
Deformation retraction      22 23
Degree of a closed path      98
Degree of a map $S^1 \to S^1$      58
Degree, local      111
Differential form      105
Differential form, closed      107
Differential form, exact      106
Differential of a function      106
Direction field      199
Distinguished neighborhood      63 68 125
Equivalence relation      28
Euler characteristic      180
Eversion, circle      104
Eversion, curve      103
Eversion, sphere      104
Fibration      see locally trivial
Fibration fundamental group      25 33
Fibration fundamental group of a polyhedron      49
Fibration fundamental group of a surface      176
Fibration fundamental group of SO(n)      74 77
Fibration fundamental group of SU(n)      84
Fibration fundamental group of the circle      53 57
Fibration fundamental group of the complex projective space      71
Fibration fundamental group of the cylinder      58
Fibration fundamental group of the real projective space      66
Fibration fundamental group of the torus      57
Fibration fundamental group of U(n)      84
Fibration fundamental group, abelian      34 37 39 49
Fibration fundamental group, induced homomorphism      34
Fundamental grupoid      32
Graph      17
Graph, circuit of a      18
Graph, connected      18
Graph, edge of a      17
Graph, edge path      18
Graph, finite      18
Graph, free vertex of a      18
Graph, maximal tree of a      23
Graph, subgraph      18
Graph, tree      17
Graph, vertex of a      17
Group      28
Group of homeomorphisms      129
Group of spinors      173
Group, action of a      154
Group, conjugation of a      151
Group, isotropy      155
Group, monodromy      156
Group, properly discontinuous      130
Group, rotation      73
Group, simplectic      91
Group, special unitary      82
Group, topological      76
Group, unitary      82 84
Hermitian inner product      82
Homotopic maps      3
Homotopic paths      26
Homotopy      3
Homotopy of pairs      20
Homotopy, class      6 28
Homotopy, equivalence      10
Homotopy, free      28
Homotopy, inverse      10
Homotopy, linear      4
Homotopy, map extension and      13
Homotopy, one-parameter family and      6
Homotopy, path      25 26
Homotopy, regular      99 100
Homotopy, relative      26
Homotopy, relative to a subspace      20
Homotopy, type      10
Hopf fibration      78
Immersion      99
Isomorphic coverings      162
Isomorphism, orientation preserving      184
Isomorphism, positive      184
Isotopy      21
Klein bottle      169 192
Lifting      55 64 122
Local homeomorphism      119
Local homomorphism      143
Locally trivial fibration      68
Locally trivial fibration, complex projective space      69
Locally trivial fibration, distinguished neighborhood      68
Locally trivial fibration, global trivialization      69
Locally trivial fibration, hopf fibration      78
Locally trivial fibration, local trivialization      68
Locally trivial fibration, path lifting      70
Locally trivial fibration, product fibration      69
Locally trivial fibration, trivial fibration      69
Manifold, Banach      147
Manifold, coherent atlas      186
Manifold, compatible coordinate systems      186
Manifold, differentiable      185
Manifold, induced orientation      186
Manifold, maximal atlas      186
Manifold, opposite orientation of a      185
Manifold, orientable      185
Manifold, orientation of a      185
Manifold, oriented      185
Manifold, Riemannian      146
Map, analytic continuation of a      124
Map, closed      127 201
Map, covering      125
Map, exponential      53
Map, extension      13
Map, inclusion      10
Map, k-th root of a      159
Map, lifting of a      122
Map, logarithm of a      158
Map, odd      61 72
Map, passing to the quotient      62
Map, proper      127 201
Map, quotient      119
Map, retraction      14
Map, section of a      123
Map, stereographic projection      41
Moebius band      188
normalizer      167
One-parameter family      3
Opposite points      65
Orbit of a point      130
Orientation of a manifold      185
Orientation of a plane curve      103
Orientation, induced      186
Orientation, product      188
Orthogonal couple      114
Pair of topological spaces      20
Parametrization, negative      31
Parametrization, positive      31
Parametrization, trivial      31
Path      25
Path lifting property      134
Path, angle function of a      55
Path, closed      26
Path, decomposition      32
Path, degree of a      53 55
Path, disorienting      197
Path, inverse of a      29
Path, lifting      64 70
Path, lifting of a      63
Path, lifting property      55
Path, operation with      28
Path, orienting      197
Path, parametrization      25
Path, partial      32
Path, piecewise $C^1$      106
Path, product of      28
Path, rectilinear      145
Path, regular      99
Path, reparametrization      25
Path, winding number of a      96
Paths, nomotopic      26
Peano curve      42
Quaternion      75
Quaternion, conjugate of a      75
Quaternion, modulus of a      75
Quaternion, orthogonal      91
Quaternion, unit      75
Quotient space, fundamental property      62 68
Radial projection      7 8 10 96
Real projective space, fundamental group of      66
Real projective space, metric on the      64
Real projective space, orientation of      189
Regular curve      99
Regular homotopy      100
Regular path      99
Regular, homotopy      99
Reparametrization of a regular curve      99
retract      14
Retract, Euclidean neighborhood      15
Rotation number      99 103
Second homotopy group      92
Simplectic group      91
Simply connected space      40
SO(N)      73 74 77
Solid torus      57
Sp(N)      91
Space      figure 8 99 103 114 139 153 154 170 179 180
Space filling curve      42
Space of type ENR      15
Space, contractible      11
Space, convex      12
Space, covering      125
Space, covering of a      125
Space, lens      170
Space, real projective      62 189
Space, semi-locally simply connected      142
Space, star      12 47
Special unitary group      82
Sphere eversion      104
Stabilizer      155
Stereographic projection      41
SU(N)      82 84
Subgroup, conjugate      151
Subgroup, normal      151
Theorem, Borsuk      16
Theorem, Borsuk — Ulam      61
Theorem, Brower fixed point      60
Theorem, Fundamental theorem of Algebra      98
Theorem, Gram — Schmidt orthonormalization      23 81 83
Theorem, Graustein — Whitney      98 101 104
Theorem, Hahn — Mazurkiewickz      42
Theorem, Hopf      103
Theorem, Koebe uniformization      178
Theorem, Mean Value Theorem      146
Theorem, Poincare      80
Theorem, Poincare — Bendixon      61
Theorem, Poincare — Bohl      4
Theorem, rank      77
Theorem, Rouche      96 98
Theorem, Sard's      43 86
Theorem, Seifert — Van Kampen      46
Theorem, Tietze — Urisohn      13
Theorem, Umlaufsatz      103
Topological group      48
Topological space, base point of a      33
Topological space, multiplication in a      47
Topological space, simply connected      40
Topology, compact-open      6
Topology, quotient      62 67
Topology, uniform convergence      6
Torus      10 28
TREE      see graph
Typical fiber      68
U(n)      82
Unique path lifting property      55 135
Unitary group      82 84
Urysohn function      17
Vector field on spheres      8
Vector field, index of a      115
Vector field, singularity of a      115
Vector space, basis of a      183
Vector space, equally oriented bases      183
Vector space, negative basis of a      184
Vector space, negative isomorphism of a      184
Vector space, opposite orientation      184
Vector space, orientation of a      183
Vector space, oriented      184
Vector space, positive basis of a      184
Vector space, positive isomorphism of a      184
Vector space, transition matrix      183
Winding number      105
Zorn Lemma      2
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