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Moise E.E. — Geometric topology in dimensions 2 and 3
Moise E.E. — Geometric topology in dimensions 2 and 3



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Название: Geometric topology in dimensions 2 and 3

Автор: Moise E.E.

Аннотация:

Geometric topology may roughly be described as the branch of the topology of manifolds which deals with questions of the existence of homeomorphisms. Only in fairly recent years has this sort of topology achieved a sufficiently high development to be given a name, but its beginnings are easy to identify. The first classic result was the Schonflies theorem (1910), which asserts that every 1-sphere in the plane is the boundary of a 2-cell.
In the next few decades, the most notable affirmative results were the "Schonflies theorem" for polyhedral 2-spheres in space, proved by J. W. Alexander, and the triangulation theorem for 2-manifolds, proved by T. Rado. But the most striking results of the 1920s were negative. In 1921 Louis Antoine published an extraordinary paper in which he showed that a variety of plausible conjectures in the topology of 3-space were false. Thus, a (topological) Cantor set in 3-space need not have a simply connected complement; therefore a Cantor set can be imbedded in 3-space in at least two essentially different ways; a topological 2-sphere in 3-space need not be the boundary of a 3-cell; given two disjoint 2-spheres in 3-space, there is not necessarily any third 2-sphere which separates them from one another in 3-space; and so on and on. The well-known "horned sphere" of Alexander (A2J appeared soon thereafter. Much later, in 1948, ihese results were extended and refined (and in some cases redone) by Ralph H. Fox and Emil Artin...


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1977

Количество страниц: 271

Добавлена в каталог: 30.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abstract complex      53
Acyclic linear graph      156
Alexander, J.W.      81 117 133
Alphabet      107
Annular strip      157
Annulus (=k-annulus)      91
Approximation (=$\phi$-approximation)      46
Approximation (=$\varepsilon$-approximation)      46
ARC      4
Arc between P and Q      19
Arcwise accessible      70
Artin, Emil      139
Ball      16
Barycenter      45
Barycentric coordinates in a Euclidean complex      3 7
Barycentric coordinates in a PL complex      54 55
Barycentric subdivision      45
Base point      97
BD      4
Betti group      162
Betti number      162
Bi-collar neighborhood      192
Bijective function      3
Bing, R.H.      252 254
Boundary      4
Brick decomposition      34
Broken line      12
Broken-line-wise connected      12
Brouwer, L.E.J.      5
Canonical homomorphism      99
Cantor set      83
Carries      198
cell      4
Cell-complex      117
Cell-decomposition      118
Cellular homeomorphism      197
Center of a pseudo-cell      223
Circular solid torus      127
Classification of 2-manifolds      155 163
Codomain of a function      3
Collar neighborhood      192
Combinatorial boundary $\partial K$ of a triangulated manifold K with boundary      5
Combinatorial equivalent      4
Combinatorial n-cell      4
Combinatorial n-manifold      5
Combinatorial solid torus (CST)      178
Commutator subgroup      99 100
Complete in the sense of Dedekind      14
COMPLEX      2
Component C(M,P) of a set M      12
Connected complex      9
Connected set      10
Connected space      10
Continuous function      4
Continuum      85
Convex      2
Convex hull      2
Coordinate mapping      54
Covering      174
Covering spaces      174
Crossing word      105
CST      178
Cyclic order on a 1-sphere      23
Cylindrical diagram of a solid torus      179
Dehn lemma      190 199
Dehn, Max      190
Dense in      4
Diagram of a Euclidean complex      52
Diagram of a link      101
Diameter $\delta M$ of a set M      34
Dimension of an abstract complex      53
Disk      4
Distance function      1
Distance functions      6
Domain of a function      3
Dual cell      223
End-point of a linear graph      73
End-point of an arc      19
Equivalent coverings      180
Euclidean complex      2
Euclidean realization of an abstract complex      54
Euler characteristic      147
Exterior of a connected 2-manifold in $R^{3}$      195
Exterior of a polygon in a plane      18
Face      2
Finite polyhedron      3
Finite-dimensional      53
Fox, Ralph H.      139
FR      4
Frame of M      72 93
Free group F(A) with alphabet A      108
Free group on n generators      108
Free simplex      27 121
Frontier      4
Full Klein bottle      179
Function      3
Fundamental group      97
General position      2
Generator of a free group      108
Generator word      104
Graeub, W.      117
Handle      151 154
Harrold, O.G., Jr.      146
Hauptvermutung, for 2-manifolds      64
Hauptvermutung, for 3-manifolds      253
Hausdorff space      6
Homeomorphic      4
Homeomorphism      4
Homogeneous space      83
Homology group $H_{n}(K)$      6
Homotopic      81
Homotopy between $f_{0}$ and $f_{1}$      81
Hudson, J.F.P.      57
Hyperplane      2
i-skeleton of a Euclidean complex      3
i-skeleton of an abstract complex      53
Imbedding      51
Induced homomorphism      98 176
Injective function      3
Inseparable in M      83
int      4
Interior of a connected 2-manifold in $R^{3}$      195
Interior of a manifold with boundary      4
Interior of a polygon in a plane      18
Interior of a set in a topological space      5
Invariance of domain      5
Irreducible continuum between P and Q      89
Isomorphism, between two abstract complexes      53
Isomorphism, between two Euclidean complexes      4
Isotopy      81
Johansson, I.      190
Join of A and $\upsilon$      44
Join of A and B      44
Jordan curve theorem      31
k-annulus      91
k-face      2
K-fold covering      177
k-sheeted covering      177
Klein bottle      151
Kline, J.R.      96
Kneser, H.      190
Latitudinal polygon      202
Lens space      211
Linear function      3
Linear graph      22
Linear ordering      14
Linearly accessible from U      29 66
Linearly ordered space      23
Link $L(\upsilon)$ of a vertex in a complex      5
Link in $R^{3}$      101
Locally commutative at P      137
Locally commutative at P, connected      89
Locally commutative at P, simply connected      142
Locally commutative at P, tame      145 254
Loop      182
Loop theorem      183 193 194
Manifold      4
Manifold with boundary      4
Mapping of one set into another      9
Mapping of one space into another      4
Mesh $\|C\|$ of a collection of sets      35 59
Metric space      1
Middle-third Cantor Set      80
Moebius band      151
Moore Kline theorem      96
n-cell      4
n-manifold      4
Neighborhood      1
Neighborhood of a set M      2
Nonsingular      182
Norm $\|P\|$ of a point P of $R^{n}$      58
Open 2-cell      223
Open cell-complex      147
Open cell-decomposition      147
Order of a vertex      24
Orientable      153 170
Orientation      153 170
Papakyriakopoulos, C.D.      183 190 193
Path      9
Path from P to Q      9
Path in M      9
Pathwise connected      9
Piecewise linear mapping      4
PL      42
PL cell-complex      118
PL complex      56
PL manifold      197
PL Schoenflies theorem      122
PL simplex      55
PLH      42
Poincare conjecture      208
Poincare, Henri      207
Polyhedron      3
Projective plane      152 164
Pseudo-cell      223
Pseudo-manifold      173
Push property      30 118
Refinement of a collection of sets      3
Regular neighborhood      155
Relation      108
retract      39
Retraction      39
Rudin, Mary Ellen      125
Schoenflies theorem      68 71 72
Semi-locally tame      145 254
Separable space      4
Separated sets      10
Separates P from Q      22
Separates two sets from one another in a connected set      22 214
Shalen, Peter B.      218 252
Shapiro, Arnold      190
Simplex in $R^{n}$      2
Simplicial mapping, for Euclidean complexes      3
Simplicial mapping, for PL complexes      56
Simply connected      98
Simply imbedded      120
Singular 2-cell      182
Singular 2-cell, with no singularities on its boundary      190
Skew graph of type 1      23 79
Skew graph of type 2      24 79
Solid torus      127
Sphere      16
Spherical shell      215
Splitting disk      223
Splitting operation      150
STABLE homeomorphism      82
Stallings, John      183 190 211
Standard n-ball $B^{n}$      16
Standard n-sphere $S^{n}$      16
Standard position      202
Star of a set in a collection of sets      89
Star of a vertex in a complex      5
Strip neighborhood      16
Strongly positive      46
Subdivision of a Euclidean complex      3
Subdivision of a PL complex      56
subspace      2
Subspace topology      2
Support      55
Surjective function      3
Tame      72
Tamely imbedded      72
Topological linear graph      22
Topological space      2
Topological tree      24
Topology induced by a neighborhood system      2
Toroidal shell      216
Torsion group      162
Torus      151
Totally disconnected set      83
TREE      24
Trefoil knot      114
Triangulated manifold (or manifold with boundary)      5
Triangulation of a set      16
Tube      223
Twisted strip      157
Two-sided surface      191
Two-sphere, with n cross-caps      152
Two-sphere, with n handles      152
Two-sphere, with n holes      152
Universal covering      175
Unknotted in C      134
Vertex      2
Vertex of order n      24
Whitehead, J.H.C.      190
Wild      72
Wild 2-spheres      132—145
Wild arcs      132—139
Wilder, R.L.      139
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