792:
points to the curve. Each configuration divide the plane into the exterior of the large circle, the interior of the Jordan curve and the region between the two into two bounded regions bounded by Jordan curves (formed of two radii, a semicircle, and one of the halves of the Jordan curve). Take the identity homeomorphism of the large circle; piecewise linear homeomorphisms between the two pairs of radii; and a homeomorphism between the two pairs of halves of the Jordan curves given by a linear reparametrization. The 4 homeomorphisms patch together on the boundary arcs to yield a homeomorphism of the plane given by the identity off the large circle and carrying one Jordan curve onto the other.
292:, defined in the smooth case by the field of unit normal vectors to the curve or in the polygonal case by points at a distance of less than Δ from the curve. In a neighbourhood of a differentiable point on the curve, there is a coordinate change in which the curve becomes the diameter of an open disk. Taking a point not on the curve, a straight line aimed at the curve starting at the point will eventually meet the tubular neighborhood; the path can be continued next to the curve until it meets the disk. It will meet it on one side or the other. This proves that the complement of the curve has at most two connected components. On the other hand, using the
380:. So dividing the circle up into small enough intervals, there are points on the curve such that the line segments joining adjacent points lie close to the curve, say by Δ. Together these line segments form a polygonal curve. If it has self-intersections, these must also create polygonal loops. Erasing these loops, results in a polygonal curve without self-intersections which still lies close to the curve; some of its vertices might not lie on the curve, but they all lie within a neighbourhood of the curve. The polygonal curve divides the plane into two regions, one bounded region
814:. (Such diffeomorphisms will be holomorphic on the interior and exterior of the curve; more general diffeomorphisms can be constructed more easily using vector fields and flows.) Regarding the smooth curve as lying inside the extended plane or 2-sphere, these analytic methods produce smooth maps up to the boundary between the closure of the interior/exterior of the smooth curve and those of the unit circle. The two identifications of the smooth curve and the unit circle will differ by a diffeomorphism of the unit circle. On the other hand, a diffeomorphism
1008:. Indeed, the smooth curve divides the 2-sphere into two parts. By the classification each is diffeomorphic to the unit disk andâtaking into account the isotopy theoremâthey are glued together by a diffeomorphism of the boundary. By the Alexander trick, such a diffeomorphism extends to the disk itself. Thus there is a diffeomorphism of the 2-sphere carrying the smooth curve onto the unit circle.
405:
396:âȘ â are continuous images of the closed unit disk. Since the original curve is contained within a small neighbourhood of the polygonal curve, the union of the images of slightly smaller concentric open disks entirely misses the original curve and their union excludes a small neighbourhood of the curve. One of the images is a bounded open set consisting of points around which the curve has
300:, it can be seen that the winding number is constant on connected components of the complement of the curve, is zero near infinity and increases by 1 when crossing the curve. Hence the curve separates the plane into exactly two components, its "interior" and its "exterior", the latter being unbounded. The same argument works for a piecewise differentiable Jordan curve.
317:
fixing the edges of the diamond, but moving one diagonal into a V shape. Compositions of homeomorphisms of this kind give rise to piecewise linear homeomorphisms of compact support; they fix the outside of a polygon and act in an affine way on a triangulation of the interior. A simple inductive argument shows that it is always possible to remove a
992:. Composing one of the diffeomorphisms with the Alexander extension allows the two diffeomorphisms to be patched together to give a homeomorphism of the 2-sphere which restricts to a diffeomorphism on the closed unit disk and the closures of its complement which it carries onto the interior and exterior of the original smooth curve. By the
413:
788:
of its interior. This is a consequence of the
Alexander trick. (The Alexander trick also establishes a homeomorphism between the solid triangle and the closed disk: the homeomorphism is just the natural radial extension of the projection of the triangle onto its circumcircle with respect to its circumcentre.)
787:
The fourth step is to prove that any homeomorphism between Jordan curves can be extended to a homeomorphism between the closures of their interiors. By the result of the third step, it is sufficient to show that any homeomorphism of the boundary of a triangle extends to a homeomorphism of the closure
321:
triangleâone for which the intersection with the boundary is a connected set made up of one or two edgesâleaving a simple closed Jordan polygon. The special homeomorphisms described above or their inverses provide piecewise linear homeomorphisms which carry the interior of the larger polygon onto the
1011:
On the other hand, the diffeomorphism can also be constructed directly using the Jordan-Schoenflies theorem for polygons and elementary methods from differential topology, namely flows defined by vector fields. When the Jordan curve is smooth (parametrized by arc length) the unit normal vectors give
400:
one; the other is an unbounded open set consisting of points of winding number zero. Repeating for a sequence of values of Δ tending to 0, leads to a union of open path-connected bounded sets of points of winding number one and a union of open path-connected unbounded sets of winding number zero. By
375:
The continuous case can also be deduced from the polygonal case by approximating the continuous curve by a polygon. The Jordan curve theorem is first deduced by this method. The Jordan curve is given by a continuous function on the unit circle. It and the inverse function from its image back to the
329:
As a corollary, it follows that any homeomorphism between simple closed polygonal curves extends to a homeomorphism between their interiors. For each polygon there is a homeomorphism of a given triangle onto the closure of their interior. The three homeomorphisms yield a single homeomorphism of the
316:
The interior of the polygon can be triangulated by small triangles, so that the edges of the polygon form edges of some of the small triangles. Piecewise linear homeomorphisms can be made up from special homeomorphisms obtained by removing a diamond from the plane and taking a piecewise affine map,
791:
The final step is to prove that given two Jordan curves there is a homeomorphism of the plane of compact support carrying one curve onto the other. In fact each Jordan curve lies inside the same large circle and in the interior of each large circle there are radii joining two diagonally opposite
428:
from the inside of the curve, i.e. they are at the end of a line segment lying entirely in the interior of the curve. In fact, a given point on the curve is arbitrarily close to some point in the interior and there is a smallest closed disk about that point which intersects the curve only on its
1029:
of the curve. Take a polygonal curve in the interior of the curve close to the boundary and transverse to the curve (at the vertices the vector field should be strictly within the angle formed by the edges). By the piecewise linear JordanâSchoenflies theorem, there is a piecewise linear
681:
can be extended to homeomorphisms between the different polygons, agreeing on common edges (closed intervals on line segments or radii). By the polygonal Jordan-Schoenflies theorem, each of these homeomorphisms extends to the interior of the polygon. Together they yield a homeomorphism
355:
between the interior of a simple Jordan curve and the open unit disk extends continuously to a homeomorphism between their closures, mapping the Jordan curve homeomorphically onto the unit circle. To prove the theorem, Carathéodory's theorem can be applied to the two regions on the
1076:
the vector field can be taken to be the standard radial vector field. Similarly the same procedure can be applied to the outside of the smooth curve, after applying Möbius transformation to map it into the finite part of the plane and â to 0. In this case the neighbourhoods
230:. The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity map off some compact set; the case of a continuous curve is then deduced by approximating by polygons. The theorem is also an immediate consequence of
488:. The size of the tiles can be taken arbitrarily small. Take the union of all the closed tiles containing at least one point of the Jordan curve. Its boundary is made up of disjoint polygonal curves. If the size of the tiles is sufficiently small, the endpoints
1259:
therefore carries the smooth curve onto this small circle. A scaling transformation, fixing 0 and â, then carries the small circle onto the unit circle. Composing these diffeomorphisms gives a diffeomorphism carrying the smooth curve onto the unit circle.
325:
Because the homeomorphism is obtained by composing finite many homeomorphisms of the plane of compact support, it follows that the piecewise linear homeomorphism in the statement of the piecewise linear Jordan-Schoenflies theorem has compact support.
470:
tiling by rectangles or squares with common or stretch bonds. It suffices to construct a polygonal path so that its distance to the Jordan curve is arbitrarily small. Orient the tessellation such no side of a tiles is parallel to any
996:
in differential topology, the homeomorphism can be adjusted to a diffeomorphism on the whole 2-sphere without changing it on the unit circle. This diffeomorphism then provides the smooth solution to the
Schoenflies problem.
801:
Proofs in the smooth case depend on finding a diffeomorphism between the interior/exterior of the curve and the closed unit disk (or its complement in the extended plane). This can be solved for example by using the smooth
334:
this homeomorphism can be extended to a homeomorphism of closure of interior of the triangle. Reversing this process this homeomorphism yields a homeomorphism between the closures of the interiors of the polygonal curves.
312:
states that there is a piecewise linear homeomorphism of the plane, with compact support, carrying the polygon onto a triangle and taking the interior and exterior of one onto the interior and exterior of the other.
372:. Composition with this homeomorphism will yield a pair of homeomorphisms which match on the Jordan curve and therefore define a homeomorphism of the Riemann sphere carrying the Jordan curve onto the unit circle.
1191:
958:
179:
458:
in its interior, there are disjoint polygonal curves in the interior with vertices on each of the line segments such that their distance to the original curve is arbitrarily small. This requires
322:
polygon with the free triangle removed. Iterating this process it follows that there is a piecewise linear homeomorphism of compact support carrying the original polygon onto a triangle.
129:
1199:
is a smooth vector field on the two sphere vanishing only at 0 and â. It has index 1 at 0 and -1 at â. Near 0 the vector field equals the radial vector field pointing towards 0. If α
497:
will lie in the interior of exactly one of the polygonal boundary curves. Its distance to the Jordan curve is less than twice the diameter of the tiles, so is arbitrarily small.
368:| â„ 1. The homeomorphisms from the Jordan curve to the circle will differ by a homeomorphism of the circle which can be extended to the unit disk (or its complement) by the
1249:
equal to 1 outside a small annulus near 0, the integral curves starting at points of the smooth curve will all reach smaller circle bounding the annulus at the same time
1064:
is chosen in "general position" so that it is not collinear with any of the finitely many edges in the triangulation. Translating if necessary, it can be assumed that
276:. Although they separate space into two regions, those regions are so twisted and knotted that they are not homeomorphic to the inside and outside of a normal sphere.
208:
253:. Although direct proofs are possible (starting for example from the polygonal case), existence of the diffeomorphism can also be deduced by using the smooth
1346:
The
Schoenflies problem can be posed in categories other than the topologically locally flat category, i.e. does a smoothly (piecewise-linearly) embedded (
17:
1789:
1004:. In fact it is an immediate consequence of the classification up to diffeomorphism of smooth oriented 2-manifolds with boundary, as described in
344:
231:
1030:
homeomorphism, affine on an appropriate triangulation of the interior of the polygon, taking the polygon onto a triangle. Take an interior point
462:
of the plane by uniformly small tiles such that if two tiles meet they have a side or a segment of a side in common: examples are the standard
2027:
1940:
1749:
504:
between the curve and a given triangle can be extended to a homeomorphism between the closures of their interiors. In fact take a sequence Δ
1140:
2238:
2066:
2005:
1888:
1843:
1735:
1710:
1679:
836:
134:
2132:
1926:
1038:
in the image triangle. There is a radial vector field on the image triangle, formed of straight lines pointing towards
2247:
2184:
2099:
666:
into a union of polygonal regions; similarly for radii for the corresponding points on Î divides the region between Î
1343:
for their contributions. Both the Brown and Mazur proofs are considered "elementary" and use inductive arguments.
429:
boundary; those boundary points are close to the original point on the curve and by construction are accessible.
2314:
2299:
99:
2309:
2252:
1902:
1857:
1727:
360:
defined by the Jordan curve. This will result in homeomorphisms between their closures and the closed disks |
401:
construction these two disjoint open path-connected sets fill out the complement of the curve in the plane.
273:
2289:
1042:. This gives a series of lines in the small triangles making up the polygon. Each defines a vector field
2083:
1978:
1819:
51:
1763:
2304:
1370: â„ 5 the question in the smooth category has an affirmative answer, and follows from the
2294:
803:
352:
293:
254:
235:
576:) on the triangle. Fix an origin in the triangle Î and scale the triangle to get a smaller one Î
1758:
463:
2198:
2147:
1020:
1001:
658:
and Î. The line segments for the accessible points on Î divide the polygonal region between Î
289:
250:
1060:
of the closure of the triangle. Each vector field is transverse to the sides, provided that
2120:
2108:
2050:
2015:
1963:
1910:
1812:
1780:
1677:
Bell, Steven R.; Krantz, Steven G. (1987), "Smoothness to the boundary of conformal maps",
377:
285:
47:
184:
8:
1921:, Translations of Mathematical Monographs, vol. 208, American Mathematical Society,
642:
with a new set of points on the Jordan curve. This will produce a second polygonal path Î
420:
Given the Jordan curve theorem, the Jordan-Schoenflies theorem can be proved as follows.
78:
74:
2269:
2215:
2164:
2115:, Studia Mathematica/Mathematische LehrbĂŒcher, vol. 15, Vandenhoeck & Ruprecht
1116:
43:
1803:
2234:
2226:
2180:
2168:
2128:
2095:
2062:
2001:
1992:, Graduate texts in mathematics, vol. 47, New York-Heidelberg: Springer-Verlag,
1922:
1884:
1839:
1831:
1731:
1706:
807:
348:
2041:
1954:
1787:
Cairns, Stewart S. (1951), "An
Elementary Proof of the Jordan-Schoenflies Theorem",
1772:
408:
Hexagonal tessellation of the plane: if 2 hexagons meet they must have a common edge
73:
The original formulation of the
Schoenflies problem states that not only does every
2261:
2207:
2156:
2036:
1993:
1949:
1798:
1768:
1688:
1315:
way (that is, the embedding extends to that of a thickened sphere), then the pair (
59:
529:
apart. Make the construction of the second step with tiles of diameter less than Δ
2046:
2011:
1959:
1906:
1808:
1776:
827:
369:
331:
258:
1985:
1363:
1238:
811:
397:
357:
297:
269:
246:
55:
1997:
1034:
in one of the small triangles of the triangulation. It corresponds to a point
806:, for which a number of direct methods are available, for example through the
268:
Such a theorem is valid only in two dimensions. In three dimensions there are
2283:
2250:(1992), "The Jordan-Schoenflies Theorem and the Classification of Surfaces",
2193:
2113:
Univalent functions, with a chapter on quadratic differentials by Gerd Jensen
2022:
1883:, Student Mathematical Library, vol. 46, American Mathematical Society,
1876:
1864:
1693:
85:
and the other (the "outside") unbounded; but also that these two regions are
2233:, Applied Mathematical Sciences, vol. 115 (Second ed.), Springer,
776:
from the interior of Î onto the interior of Î. By construction it has limit
638:
between the closure of their interiors. Now carry out the same process for Δ
27:
Extends the Jordan curve theorem to characterize the inner and outer regions
1744:
1340:
1312:
1269:
459:
262:
86:
2127:, Grundlehren der Mathematischen Wissenschaften, vol. 299, Springer,
967:
is a smooth function with values in , equal to 0 near 0 and 1 near 1, and
412:
1970:
1935:
1371:
1277:
1095:
with a negative sign, pointing away from the point at infinity. Together
343:
The Jordan-Schoenflies theorem for continuous curves can be proved using
82:
31:
2273:
2219:
2160:
2075:
1881:
Lectures on
Surfaces: (Almost) Everything You Wanted to Know about Them
1719:
1245:, but not the integral curves themselves. For an appropriate choice of
432:
The second step is to prove that given finitely many accessible points
606:) and the smaller triangle. There is a piecewise linear homeomorphism
424:
The first step is to show that a dense set of points on the curve are
1210:
467:
2265:
2211:
2142:
245:
If the curve is smooth then the homeomorphism can be chosen to be a
1362: = 4, the problem is still open for both categories. See
257:
for the interior and exterior of the curve in combination with the
631:. By the Jordan-Schoenflies theorem it extends to a homeomorphism
288:
can be proved in a straightforward way. Indeed, the curve has a
1297:
1237:
a smooth positive function, changes the parametrization of the
1086:
of the triangles have negative indices. Take the vector fields
211:
90:
1703:
The Cauchy transform, potential theory, and conformal mapping
2025:(1960), "A reduction of the Schoenflies extension problem",
1268:
There does exist a higher-dimensional generalization due to
1186:{\displaystyle \displaystyle {X=\sum \psi _{i}\cdot X_{i}.}}
404:
279:
1747:(1960), "A proof of the generalized Schoenflies theorem",
613:
of the polygonal curve onto the smaller triangle carrying
1901:, Proc. Symp. Pure Math., vol. XXX, Providence, RI:
1856:, Translations of Mathematical Monographs, vol. 26,
261:
for diffeomorphisms of the circle and a result on smooth
593:
be the points at the intersection of the radius through
525:
on the Jordan curve Πwith successive points less than Δ
308:
Given a simple closed polygonal curve in the plane, the
131:
is a simple closed curve, then there is a homeomorphism
81:
separate the plane into two regions, one (the "inside")
1822:(1913), "Zur RĂ€nderzuordnung bei konformer Abbildung",
953:{\displaystyle \displaystyle {F(re^{i\theta })=r\exp,}}
820:
of the unit circle can be extended to a diffeomorphism
1115:â 0 form an open cover of the 2-sphere. Take a smooth
516:, ... decreasing to zero. Choose finitely many points
174:{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}
1144:
1143:
840:
839:
674:
into a union of polygonal regions. The homeomorphism
187:
137:
102:
1000:
The Jordan-Schoenflies theorem can be deduced using
757:
increase to the region inside Î; and the triangles Î
711:. Continuing in this way produces polygonal curves Î
1854:
Geometric theory of functions of a complex variable
2056:
1576:
1402:
1185:
952:
500:The third step is to prove that any homeomorphism
202:
173:
123:
1217:tends to +â, the flow send points to 0; while as
2281:
2080:Elements of the topology of plane sets of points
1790:Proceedings of the American Mathematical Society
1072:are at the origin 0. On the triangle containing
214:in the plane. Elementary proofs can be found in
1875:
1547:
1435:
2231:Partial differential equations I. Basic theory
2057:Napier, Terrence; Ramachandran, Mohan (2011),
1705:, Studies in Advanced Mathematics, CRC Press,
249:. Proofs in this case rely on techniques from
2028:Bulletin of the American Mathematical Society
1941:Bulletin of the American Mathematical Society
1750:Bulletin of the American Mathematical Society
1836:Differential geometry of curves and surfaces
1818:
1494:
1221:tends to ââ points are sent to â. Replacing
650:and Î. There is likewise a second triangle Î
2140:
1899:A Monge-Ampére equation in complex analysis
1726:, Colloquium Publications -, vol. 40,
310:piecewise linear JordanâSchoenflies theorem
2143:"Beitrage zur Theorie der Punktmengen III"
2119:
2107:
2089:
1676:
1660:
1591:
1505:
1417:
1354:-sphere bound a smooth (piecewise-linear)
239:
2246:
2196:(1961), "On gradient dynamical systems",
2040:
1953:
1916:
1802:
1762:
1692:
1655:
732:between the closures of their interiors;
542:to be the points on the polygonal curve Î
227:
161:
146:
124:{\displaystyle C\subset \mathbb {R} ^{2}}
111:
1990:Geometric topology in dimensions 2 and 3
1830:
441:on the curve connected to line segments
416:A standard brickwork tiling of the plane
411:
403:
280:Proofs of the JordanâSchoenflies theorem
89:to the inside and outside of a standard
2174:
1896:
1851:
1650:
1618:
1586:
1499:
1412:
1397:
1288:, which is also called the generalized
772:patch together to give a homeomorphism
68:
14:
2282:
2225:
2074:
1969:
1863:
1786:
1645:
1640:
1612:
1581:
1407:
1392:
1339:-sphere. Brown and Mazur received the
1005:
780:on the boundary curves Î and Î. Hence
219:
215:
2192:
2021:
1984:
1934:
1743:
1724:The Geometric Topology of 3-Manifolds
1680:Rocky Mountain Journal of Mathematics
1635:
1521:
1467:
1449:
1285:
1281:
1273:
693:onto the closure of the interior of Î
223:
2125:Boundary behaviour of conformal maps
1938:(1959), "On embeddings of spheres",
1718:
1700:
1596:
1561:
1542:
1526:
1479:
1454:
1422:
1350: − 1)-sphere in the
284:For smooth or polygonal curves, the
96:An alternative statement is that if
2059:An Introduction to Riemann Surfaces
1975:Lectures on the h-cobordism theorem
689:of the closure of the interior of Î
338:
24:
1296: − 1)-dimensional
1263:
763:increase to Î. The homeomorphisms
303:
25:
18:Generalized Schoenflies conjecture
2326:
2177:Elements of differential topology
1804:10.1090/S0002-9939-1951-0046635-9
330:boundary of the triangle. By the
2076:Newman, Maxwell Herman Alexander
584:from the original triangle. Let
232:Carathéodory's extension theorem
2042:10.1090/S0002-9904-1960-10420-X
1955:10.1090/S0002-9904-1959-10274-3
1919:An introduction to Morse theory
1773:10.1090/S0002-9904-1960-10400-4
1626:
1603:
1567:
1323:) is homeomorphic to the pair (
796:
62:it is often referred to as the
1577:Napier & Ramachandran 2011
1555:
1533:
1512:
1485:
1473:
1461:
1440:
1429:
1403:Napier & Ramachandran 2011
1383:
1205:is the smooth flow defined by
942:
936:
933:
927:
915:
906:
900:
894:
888:
879:
864:
845:
784:is the required homeomorphism.
197:
191:
156:
13:
1:
2253:American Mathematical Monthly
2092:An invitation to Morse theory
1903:American Mathematical Society
1879:; Climenhaga, Vaughn (2008),
1858:American Mathematical Society
1852:Goluzin, GennadiÄ M. (1969),
1728:American Mathematical Society
1670:
1012:a non-vanishing vector field
1621:, p. 173, Theorem 6.4.3
1213:and â a repelling point. As
265:from differential topology.
7:
1548:Katok & Climenhaga 2008
1436:Katok & Climenhaga 2008
64:JordanâSchoenflies theorem.
10:
2331:
2175:Shastri, Anant R. (2011),
2094:(2nd ed.), Springer,
2090:Nicolaescu, Liviu (2011),
2084:Cambridge University Press
1979:Princeton University Press
1897:Kerzman, Norberto (1977),
1615:, p. 182, Theorem 1.9
751:. The regions inside the Î
1998:10.1007/978-1-4612-9906-6
1917:Matsumoto, Yukio (2002),
1292:. It states that, if an (
1125:subordinate to the cover
580:at a distance less than Δ
384:and one unbounded region
274:Alexander's horned sphere
2141:Schoenflies, A. (1906),
1871:(2nd ed.), Springer
1820:Carathéodory, Constantin
1701:Bell, Steven R. (1992),
1694:10.1216/rmj-1987-17-1-23
1377:
826:of the unit disk by the
804:Riemann mapping theorem
723:with a homomeomorphism
294:Cauchy integral formula
255:Riemann mapping theorem
46:is a sharpening of the
1592:Bell & Krantz 1987
1418:Bell & Krantz 1987
1335:is the equator of the
1253:. The diffeomorphism α
1187:
954:
464:hexagonal tessellation
417:
409:
345:Carathéodory's theorem
204:
175:
125:
2315:Mathematical problems
2300:Differential topology
2199:Annals of Mathematics
2148:Mathematische Annalen
2121:Pommerenke, Christian
2109:Pommerenke, Christian
1869:Differential topology
1832:do Carmo, Manfredo P.
1824:Göttingen Nachrichten
1303:is embedded into the
1188:
1021:tubular neighbourhood
1002:differential topology
955:
415:
407:
351:. It states that the
290:tubular neighbourhood
251:differential topology
205:
176:
126:
2310:Theorems in topology
1307:-dimensional sphere
1276:) and independently
1209:, the point 0 is an
1141:
837:
378:uniformly continuous
286:Jordan curve theorem
203:{\displaystyle f(C)}
185:
135:
100:
69:Original formulation
48:Jordan curve theorem
1290:Schoenflies theorem
1051:on a neighbourhood
828:Alexander extension
75:simple closed curve
40:Schoenflies theorem
36:Schoenflies problem
2290:Geometric topology
2248:Thomassen, Carsten
2227:Taylor, Michael E.
2161:10.1007/bf01449982
1183:
1182:
1117:partition of unity
950:
949:
563:. Take the points
466:; or the standard
418:
410:
238:, as discussed in
236:conformal mappings
200:
171:
121:
52:Arthur Schoenflies
44:geometric topology
2240:978-1-4419-7054-1
2068:978-0-8176-4692-9
2007:978-0-387-90220-3
1890:978-0-8218-4679-7
1877:Katok, Anatole B.
1845:978-0-13-212589-5
1838:, Prentice-Hall,
1737:978-0-8218-1040-8
1712:978-0-8493-8270-3
1495:Carathéodory 1913
808:Dirichlet problem
349:conformal mapping
16:(Redirected from
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2276:
2243:
2222:
2189:
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2137:
2116:
2104:
2086:
2071:
2053:
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2018:
1981:
1966:
1957:
1931:
1913:
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1740:
1715:
1697:
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1630:
1624:
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1601:
1571:
1565:
1564:, pp. 29â32
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1477:
1471:
1470:, pp. 26â29
1465:
1459:
1444:
1438:
1433:
1427:
1387:
1270:Morton Brown
1211:attracting point
1192:
1190:
1189:
1184:
1181:
1177:
1176:
1164:
1163:
991:
980:
966:
959:
957:
956:
951:
948:
863:
862:
825:
819:
810:on the curve or
376:unit circle are
339:Continuous curve
240:Pommerenke (1992
228:Thomassen (1992)
209:
207:
206:
201:
180:
178:
177:
172:
170:
169:
164:
155:
154:
149:
130:
128:
127:
122:
120:
119:
114:
21:
2330:
2329:
2325:
2324:
2323:
2321:
2320:
2319:
2305:Diffeomorphisms
2280:
2279:
2266:10.2307/2324180
2241:
2212:10.2307/1970311
2187:
2135:
2102:
2069:
2008:
1986:Moise, Edwin E.
1929:
1891:
1846:
1764:10.1.1.228.5491
1738:
1713:
1673:
1668:
1661:Nicolaescu 2011
1631:
1627:
1608:
1604:
1572:
1568:
1562:Bing & 1983
1560:
1556:
1538:
1534:
1517:
1513:
1506:Pommerenke 1975
1490:
1486:
1478:
1474:
1466:
1462:
1445:
1441:
1434:
1430:
1388:
1384:
1380:
1278:Barry Mazur
1266:
1264:Generalizations
1258:
1239:integral curves
1204:
1172:
1168:
1159:
1155:
1145:
1142:
1139:
1138:
1133:
1124:
1110:
1101:
1094:
1085:
1059:
1050:
1028:
1018:
994:isotopy theorem
982:
968:
964:
855:
851:
841:
838:
835:
834:
821:
815:
812:Bergman kernels
799:
771:
762:
756:
750:
740:
731:
722:
717:and triangles Î
716:
710:
703:
696:
692:
688:
680:
673:
669:
665:
661:
657:
653:
649:
645:
641:
637:
630:
621:
612:
605:
592:
583:
579:
575:
562:
554:
545:
541:
532:
528:
524:
515:
511:
507:
496:
487:
479:
457:
449:
440:
370:Alexander trick
353:Riemann mapping
341:
332:Alexander trick
306:
304:Polygonal curve
282:
270:counterexamples
259:Alexander trick
242:, p. 25).
186:
183:
182:
165:
160:
159:
150:
145:
144:
136:
133:
132:
115:
110:
109:
101:
98:
97:
71:
28:
23:
22:
15:
12:
11:
5:
2328:
2318:
2317:
2312:
2307:
2302:
2297:
2295:Homeomorphisms
2292:
2278:
2277:
2260:(2): 116â130,
2244:
2239:
2223:
2206:(1): 199â206,
2194:Smale, Stephen
2190:
2185:
2172:
2155:(2): 286â328,
2138:
2134:978-3540547518
2133:
2117:
2105:
2100:
2087:
2072:
2067:
2054:
2035:(2): 113â115,
2023:Morse, Marston
2019:
2006:
1982:
1967:
1932:
1928:978-0821810224
1927:
1914:
1894:
1889:
1873:
1865:Hirsch, Morris
1861:
1849:
1844:
1828:
1816:
1797:(6): 860â867,
1784:
1741:
1736:
1716:
1711:
1698:
1672:
1669:
1667:
1666:
1664:
1663:
1658:
1656:Matsumoto 2002
1653:
1648:
1643:
1638:
1625:
1623:
1622:
1616:
1602:
1600:
1599:
1594:
1589:
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1579:
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1529:
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1472:
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1457:
1452:
1439:
1428:
1426:
1425:
1420:
1415:
1410:
1405:
1400:
1395:
1381:
1379:
1376:
1364:Mazur manifold
1265:
1262:
1254:
1200:
1194:
1193:
1180:
1175:
1171:
1167:
1162:
1158:
1154:
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1129:
1120:
1106:
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1081:
1055:
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920:
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911:
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896:
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887:
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881:
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530:
526:
520:
513:
509:
505:
498:
492:
483:
475:
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445:
436:
430:
398:winding number
358:Riemann sphere
340:
337:
305:
302:
298:winding number
281:
278:
247:diffeomorphism
199:
196:
193:
190:
168:
163:
158:
153:
148:
143:
140:
118:
113:
108:
105:
93:in the plane.
70:
67:
58:curves in the
26:
9:
6:
4:
3:
2:
2327:
2316:
2313:
2311:
2308:
2306:
2303:
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2293:
2291:
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2287:
2285:
2275:
2271:
2267:
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2259:
2255:
2254:
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2242:
2236:
2232:
2228:
2224:
2221:
2217:
2213:
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2201:
2200:
2195:
2191:
2188:
2186:9781439831601
2182:
2179:, CRC Press,
2178:
2173:
2170:
2166:
2162:
2158:
2154:
2150:
2149:
2144:
2139:
2136:
2130:
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2122:
2118:
2114:
2110:
2106:
2103:
2101:9781461411048
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2093:
2088:
2085:
2081:
2077:
2073:
2070:
2064:
2060:
2055:
2052:
2048:
2043:
2038:
2034:
2030:
2029:
2024:
2020:
2017:
2013:
2009:
2003:
1999:
1995:
1991:
1987:
1983:
1980:
1976:
1972:
1968:
1965:
1961:
1956:
1951:
1947:
1943:
1942:
1937:
1933:
1930:
1924:
1920:
1915:
1912:
1908:
1904:
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1892:
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1825:
1821:
1817:
1814:
1810:
1805:
1800:
1796:
1792:
1791:
1785:
1782:
1778:
1774:
1770:
1765:
1760:
1756:
1752:
1751:
1746:
1745:Brown, Morton
1742:
1739:
1733:
1729:
1725:
1721:
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1714:
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1109:
1105:
1098:
1093:
1089:
1084:
1080:
1075:
1071:
1067:
1063:
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1054:
1049:
1045:
1041:
1037:
1033:
1025:
1022:
1015:
1009:
1007:
1006:Hirsch (1994)
1003:
998:
995:
989:
985:
979:
975:
971:
945:
939:
930:
924:
921:
918:
912:
909:
903:
897:
891:
885:
882:
876:
873:
870:
867:
859:
856:
852:
848:
842:
833:
832:
831:
829:
824:
818:
813:
809:
805:
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786:
783:
779:
775:
770:
766:
761:
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748:
744:
739:
735:
730:
726:
721:
715:
707:
700:
685:
677:
634:
629:
625:
620:
616:
609:
604:
600:
596:
591:
587:
574:
570:
566:
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557:
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549:
546:intersecting
540:
536:
523:
519:
503:
499:
495:
491:
486:
482:
478:
474:
469:
465:
461:
460:tessellations
456:
452:
448:
444:
439:
435:
431:
427:
423:
422:
421:
414:
406:
402:
399:
395:
391:
387:
383:
379:
373:
371:
367:
363:
359:
354:
350:
346:
336:
333:
327:
323:
320:
314:
311:
301:
299:
295:
291:
287:
277:
275:
271:
266:
264:
260:
256:
252:
248:
243:
241:
237:
233:
229:
225:
221:
220:Cairns (1951)
217:
216:Newman (1939)
213:
194:
188:
166:
151:
141:
138:
116:
106:
103:
94:
92:
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80:
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2230:
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2197:
2176:
2152:
2146:
2124:
2112:
2091:
2079:
2061:, Springer,
2058:
2032:
2026:
1989:
1974:
1971:Milnor, John
1948:(2): 59â65,
1945:
1939:
1936:Mazur, Barry
1918:
1898:
1880:
1868:
1853:
1835:
1823:
1794:
1788:
1757:(2): 74â76,
1754:
1748:
1723:
1702:
1684:
1678:
1651:Shastri 2011
1628:
1619:Shastri 2011
1605:
1587:Kerzman 1977
1569:
1557:
1550:, Lecture 36
1535:
1514:
1502:, p. 44
1500:Goluzin 1969
1487:
1482:, p. 29
1475:
1463:
1442:
1431:
1413:Kerzman 1977
1398:Shastri 2011
1385:
1367:
1359:
1358:-ball? For
1355:
1351:
1347:
1345:
1341:Veblen Prize
1336:
1332:
1328:
1324:
1320:
1316:
1313:locally flat
1308:
1304:
1300:
1293:
1289:
1286:Morse (1960)
1267:
1255:
1250:
1246:
1242:
1234:
1230:
1226:
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797:Smooth curve
781:
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315:
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267:
244:
224:Moise (1977)
210:is the unit
95:
87:homeomorphic
72:
63:
39:
35:
29:
1720:Bing, R. H.
1646:Hirsch 1994
1641:Milnor 1965
1613:Hirsch 1994
1582:Taylor 2011
1408:Taylor 2011
1393:Hirsch 1994
1372:h-cobordism
986:(Ξ + 2Ï) =
364:| †1 and |
32:mathematics
2284:Categories
1671:References
1636:Smale 1961
1522:Moise 1977
1468:Moise 1977
1450:Moise 1977
426:accessible
181:such that
2169:123992220
1826:: 509â518
1759:CiteSeerX
1687:: 23â40,
1597:Bell 1992
1543:Bing 1983
1527:Bing 1983
1480:Bing 1983
1455:Bing 1983
1423:Bell 1992
1374:theorem.
1331:), where
1166:⋅
1157:ψ
1153:∑
940:θ
925:ψ
922:−
904:θ
886:ψ
877:
860:θ
654:between Î
646:between Î
533:and take
468:brickwork
157:→
107:⊂
2229:(2011),
2123:(1992),
2111:(1975),
2078:(1939),
1988:(1977),
1973:(1965),
1867:(1994),
1834:(1976),
1722:(1983),
1134:and set
1111:'s with
1102:and the
990:(Ξ) + 2Ï
741:extends
704:extends
296:for the
272:such as
2274:2324180
2220:1970311
2051:0117694
2016:0488059
1964:0117693
1911:0454082
1813:0046635
1781:0117695
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1284:) with
1280: (
1272: (
981:, with
388:. Both
263:isotopy
83:bounded
77:in the
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2183:
2167:
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2065:
2049:
2014:
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963:where
212:circle
91:circle
56:Jordan
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34:, the
2270:JSTOR
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2165:S2CID
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1378:Notes
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1019:in a
670:and Î
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1885:ISBN
1840:ISBN
1732:ISBN
1707:ISBN
1282:1959
1274:1960
1068:and
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392:and
319:free
234:for
226:and
2262:doi
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2157:doi
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1994:doi
1950:doi
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2082:,
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2033:66
2031:,
2012:MR
2010:,
2000:,
1977:,
1960:MR
1958:,
1946:65
1944:,
1907:MR
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218:,
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578:1
573:i
569:A
567:(
565:f
560:i
556:B
552:i
548:A
544:1
539:i
535:C
531:1
527:1
522:i
518:A
514:3
510:2
506:1
502:f
494:i
490:B
485:i
481:B
477:i
473:A
455:i
451:B
447:i
443:A
438:i
434:A
394:V
390:U
386:V
382:U
366:z
362:z
198:)
195:C
192:(
189:f
167:2
162:R
152:2
147:R
142::
139:f
117:2
112:R
104:C
20:)
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