|
Embeddings of surfaces in euclidean three-space
Author(s):
C. E.
Burgess
Journal:
Bull. Amer. Math. Soc.
81
(1975),
795-818.
MSC (1970):
Primary 55A30, 55A35, 57A10, 57A35, 57A45, 57A50, 57C55
MathSciNet review:
0375319
Retrieve article in:
PDF
References |
Similar articles |
Additional information
References:
- 1.
- J. W. Alexander, On the subdivision of 3-space by a polyhedron, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 6-8.
- 2.
- J. W. Alexander, An example of a simply connected surface bounding a region which is not simply connected, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 8-10.
- 3.
- J. W. Alexander, Remarks on a point set constructed by Antoine, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 10-12.
- 4.
- W. R. Alford, Some "nice" wild 2-spheres in E, Topology of 3-Manifolds and Related Topics (Proc. Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 29-33. MR 25 #4504. MR 141091
- 5.
- M. L. Antoine, Sur la possibilité d'étendre l'homéomorphie de deux figures a leurs voisinages, C. R. Acad. Sci. Paris 171 (1920), 661-663.
- 6.
- M. L. Antoine, Sur l'homéomorphie de deux figures et de leurs voisinages, J. Math. Pures Appl. 86 (1921), 221-325.
- 7.
- Steve Armentrout, Upper semi-continuous decompositions of E, Ann. of Math. (2) 78 (1963), 605-618. MR 27 #6255. MR 156331
- 8.
- Steve Armentrout, Concerning cellular decompositions of 3-manifolds with boundary, Trans. Amer. Math. Soc. 137 (1969), 231-236. MR 38 #5224. MR 236931
- 9.
- B. J. Ball, The sum of two solid horned spheres, Ann. of Math. (2) 69 (1959), 253-257. MR 21 #321. MR 101511
- 10.
- R. H. Bing, A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann. of Math. (2) 56 (1952), 354-362. MR 14, 192. MR 49549
- 11.
- R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145-158. MR 15, 816. MR 61377
- 12.
- R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 456-483. MR 19, 300. MR 87090
- 13.
- R. H. Bing, Conditions under which a surface in E, Fund. Math. 47 (1959), 105-139. MR 21 #5954. MR 107229
- 14.
- R. H. Bing, An alternative proof that 3-manifolds can be triangulated, Ann. of Math. (2) 69 (1959), 37-65. MR 20 #7269. MR 100841
- 15.
- R. H. Bing, A wild surface each of whose arcs is tame, Duke Math. J. 28 (1961), 1-15. MR 23 #A630. MR 123302
- 16.
- R. H. Bing, A surface is tame if its complement is 1-ULC, Trans. Amer. Math. Soc. 101 (1961), 294-305. MR24#A1117. MR 131265
- 17.
- R. H. Bing, Each disk in E, Amer. J. Math. 84 (1962), 583-590. MR 26 #4331. MR 146811
- 18.
- R. H. Bing, Each disk in E, Amer. J. Math. 84 (1962), 591-599. MR 26 #4332. MR 146812
- 19.
- R. H. Bing, Approximating surfaces from the side, Ann. of Math. (2) 77 (1963), 145-192. MR27 #731. MR 150744
- 20.
- R. H. Bing, Spheres in E, Amer. Math. Monthly 71 (1964), 353-364. MR 29 #2789. MR 165507
- 21.
- R. H. Bing, Improving the side approximation theorem, Trans. Amer. Math. Soc. 116 (1965), 511-525. MR 33 #704. MR 192479
- 22.
- R. H. Bing, Improving the intersections of lines and surfaces, Michigan Math. J. 14 (1967), 155-159. MR 34 #6743. MR 206927
- 23.
- H. G. Bothe, Ein homogen wilder Knoten, Fund. Math. 60 (1967), 271-283. MR 35 #7327. MR 216494
- 24.
- William S. Boyd, Jr., Repairing embeddings of 3-cells with monotone maps of E, Trans. Amer. Math. Soc. 161 (1971), 123-144. MR 43 #8064. MR 282352
- 25.
- Morton Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. MR 22 #8470b. MR 117695
- 26.
- Morton Brown, Locally flat imbeddings of topological manifolds, Ann. of Math. (2) 75 (1962), 331-341. MR 24 #A3637. MR 133812
- 27.
- J. L. Bryant and R. C. Lacher, Embeddings with mapping cylinder neighborhoods, Topology (to appear). MR 394680
- 28.
- C. E. Burgess, Characterizations of tame surfaces in E, Trans. Amer. Math. Soc. 114 (1965), 80-97. MR 31 #728. MR 176456
- 29.
- C. E. Burgess, Pairs of 3-cells with intersecting boundaries in E, Amer. J. Math. 88 (1966), 309-313. MR 33 #4909. MR 196723
- 30.
- C. E. Burgess, Extending self homeomorphisms of a wild sphere described by Gillman, Duke Math. J. 39 (1972), 725-728. MR 47 #2588. MR 314036
- 31.
- C. E. Burgess and J. W. Cannon, Tame subsets of spheres in E, Proc. Amer. Math. Soc. 22 (1969), 395-401. MR 39 #3469. MR 242135
- 32.
- C. E. Burgess and J. W. Cannon, Embeddings of surfaces in E, Rocky Mountain J. Math. 1 (1971), No. 2, 259-344. MR 43 #4008. MR 278277
- 33.
- C. E. Burgess and L. D. Loveland, Sequentially 1-ULC surfaces in E, Proc. Amer. Math. Soc. 19 (1968), 653-659. MR 37 #3546. MR 227962
- 34.
- J. W. Cannon, Characterization of taming sets on 2-spheres, Trans. Amer. Math. Soc. 147 (1970), 289-299. MR 41 #2644. MR 257996
- 35.
- J. W. Cannon, Sets which can be missed by side approximations to spheres, Pacific J. Math. 34 (1970), 321-334. MR 42 #2447. MR 267545
- 36.
- J. W. Cannon, *-Taming sets for crumpled cubes. I: Basic properties, Trans. Amer. Math. Soc. 161 (1971), 429-440. MR 43 #8065. MR 282353
- 37.
- J. W. Cannon, *-Taming sets for crumpled cubes. II: Horizontal sections in closed sets, Trans. Amer. Math. Soc. 161 (1971), 441-446. MR 43 #8066. MR 282354
- 38.
- J. W. Cannon, *-Taming sets for crumpled cubes. III: Horizontal sections in 2-spheres, Trans. Amer. Math. Soc. 161 (1971), 447-456. MR 43 #8067. MR 282355
- 39.
- J. W. Cannon, Characterizations of tame subsets of 2-spheres in E, Amer. J. Math. 94 (1972), 173-188. MR 45 #5980. MR 296921
- 40.
- J. W. Cannon, ULC properties in neighborhoods of embedded surfaces and curves in E, Canad. J. Math. 25 (1973), 31-73. MR 47 #2589. MR 314037
- 41.
- J. W. Cannon, New proofs of Bing's approximation theorems for surfaces, Pacific J. Math. 46 (1973), 361-379. MR 48 #7289. MR 328947
- 42.
- J. W. Cannon, Singular side approximations for 2-spheres in E, Illinois J. Math. 18 (1974), 27-36. MR 334215
- 43.
- J. W. Cannon, Piercing disks with tame arcs, Michigan Math. J. 20 (1974), 345-347. MR 341490
- 44.
- J. W. Cannon and L. D. Loveland, A 2-sphere in E, Trans. Amer. Math. Soc. 195 (1974), 345-355. MR 343273
- 45.
- L. O. Cannon, Sums of solid horned spheres, Trans. Amer. Math. Soc. 122 (1966), 203-228. MR 32 #8322. MR 190912
- 46.
- J. C. Cantrell, Almost locally fiat embeddings of S, Bull. Amer. Math. Soc. 69 (1963), 716-718. MR 27 #4237. MR 154288
- 47.
- A. V. Černavskiĭ, Coincidence of local flatness and local simple connectedness for embeddings of (n-1)-dimensional manifolds in n-dimensional manifolds, Mat. Sb. 91 (133) (1973), 279-286=Math. USSR Sb. 20 (1973), 297-304. MR 334222
- 48.
- R. J. Daverman, A new proof for the Hosay-Lininger theorem about crumpled cubes, Proc. Amer. Math. Soc. 23 (1969), 52-54. MR 39 #7578. MR 246274
- 49.
- R. J. Daverman, Slicing theorems for n-spheres in Euclidean (n+l)-space, Trans. Amer. Math. Soc. 166 (1972), 479-489. MR 45 #4422. MR 295356
- 50.
- R. J. Daverman, Locally nice codimension one manifolds are locally flat, Bull. Amer. Math. Soc. 79 (1973), 410-413. MR 47 #9628. MR 321095
- 51.
- R. J. Daverman, On the scarcity of tame disks in certain wild cells, Fund. Math. 79 (1973), 63-77.MR 48 #5085. MR 326742
- 52.
- R. J. Daverman, Approximating polyhedra in codimension one spheres embedded in S, Pacific J. Math. 51 (1974), 417-426. MR 362326
- 53.
- R. J. Daverman, Wild spheres in E, Trans. Amer. Math. Soc. 206 (1975), 347-359. MR 375329
- 54.
- R. J. Daverman, Singular regular neighborhoods and local flatness in codimension one, Proc. Amer. Math. Soc. (to appear). MR 420630
- 55.
- R. J. Daverman, Flattening a codimension one manifold by piercing with cells (to appear). MR 795714
- 56.
- R. J. Daverman, A summary of results and problems concerning flatness of Codimension one spheresin E, Geometrical Topology, Proc. Geometrical Topology Conf. (Park City, Utah, 1974), pp. 156-165. MR 407847
- 57.
- R. J. Daverman, Sewings of closed n-cell complements (to appear).
- 58.
- R. J. Daverman and W. T. Eaton, A dense set of sewings of two crumpled cubes yields S, Fund. Math. 65 (1969), 51-60. MR 39 #4821. MR 243500
- 59.
- P. H. Doyle and J. G. Hocking, Some results on tame discs and spheres in E, Proc. Amer. Math. Soc. 11 (1960), 832-836. MR 23 #A4133. MR 126839
- 60.
- W. T. Eaton, Side approximations in crumpled cubes, Duke Math. J. 35 (1968), 707-719. MR 38 #694. MR 232369
- 61.
- W. T. Eaton, Cross sectionally simple spheres, Bull. Amer. Math. Soc. 75 (1969), 375-378. MR 39 #957. MR 239600
- 62.
- W. T. Eaton, Taming a surface by piercing with disks, Proc. Amer. Math. Soc. 22 (1969), 724-727. MR 39 #7579. MR 246275
- 63.
- W. T. Eaton, The sum of solid spheres, Michigan Math. J. 19 (1972), 193-207. MR 46 #8227. MR 309116
- 64.
- W. T. Eaton, A two sided approximation theorem for 2-spheres, Pacific J. Math. 44 (1973), 461-487. MR 47 #9606. MR 321073
- 65.
- W. T. Eaton, Applications of a mismatch theorem to decomposition spaces, Fund. Math, (to appear). MR 391095
- 66.
- M. K. Fort, Jr., A wild sphere which can be pierced at each point by a straight line segment, Proc. Amer. Math. Soc. 14 (1963), 994-995. MR 28 #2528. MR 159311
- 67.
- R. H. Fox and E. Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979-990. MR 10, 317. MR 27512
- 68.
- D. S. Gillman, Tame subsets of 2-spheres in E, Topology of 3-manifolds and Related Topics (Proc. Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 26-28. MR 25 #4503.
- 69.
- D. S. Gillman, Note concerning a wild sphere of Bing, Duke Math. J. 31 (1964), 247-254. MR 28 #4525. MR 161317
- 70.
- O. G. Harrold, Jr., Locally tame curves and surfaces in three-dimensional manifolds, Bull. Amer. Math. Soc 63 (1957), 293-305. MR 19, 568. MR 88718
- 71.
- O. G. Harrold, Jr., Locally peripherally unknotted surfaces in E, Ann. of Math. (2) 69 (1959), 276-290. MR 21 #4399a. MR 105660
- 72.
- J. P. Hempel, A surface in S, Trans. Amer. Math. Soc. 111 (1964), 273-287. MR 28 #3409. MR 160195
- 73.
- David W. Henderson, Extension of Dehn's Lemma and the loop theorem, Trans. Amer. Math. Soc. 120 (1965), 448-469. MR 32 #4686. MR 187233
- 74.
- Norman Hosay, A proof of the slicing theorem for 2-spheres, Bull. Amer. Math. Soc. 75 (1969), 370-374. MR 39 #956. MR 239599
- 75.
- R. A. Jensen, Cross sectionally connected 2-spheres are tame, Bull. Amer. Math. Soc. 76 (1970), 1036-1038. MR 42 #5241. MR 270352
- 76.
- R. A. Jensen and L. D. Loveland, Surfaces of vertical order 3 are tame, Bull. Amer. Math. Soc. 76 (1970), 151-154. MR 40 #3520. MR 250281
- 77.
- S. Kinoshita and C. D. Sikkema, Penetration indices of arc and ball pairs and uncountably many quasi-translations of the 3-sphere, Michigan Math. J. 20 (1974), 309-319. MR 334216
- 78.
- R. C. Kirby, On the set of nonlocally flat points for a submanifold of codimension one, Ann. of Math. (2) 88 (1968), 281-290. MR 38 #5193. MR 236900
- 79.
- F. M. Lister, Simplifying intersections of disks in Bmg's side approximation theorem, Pacific J. Math. 22 (1967), 281-295. MR 35 #7317. MR 216484
- 80.
- F. M. Lister, Tame boundary sets of crumpled cubes in E, Proc. Amer. Math. Soc. 25 (1970), 377-378. 41 #2647. MR 257999
- 81.
- L. D. Loveland, Tame subsets of spheres in E, Pacific J. Math. 19 (1966), 489-517. MR 37 #903. MR 225309
- 82.
- L. D. Loveland, Tame surfaces and tame subsets of spheres in E, Trans. Amer. Math. Soc. 123 (1966), 355-368. MR 33 #7990. MR 199850
- 83.
- L. D. Loveland, Tameness implied by extending a homeomorphism to a point, Proc. Amer. Math. Soc. 23 (1969), 287-293. MR 40 #3521. MR 250282
- 84.
- L. D. Loveland, Vertically countable spheres and their wild sets, Pacific J. Math. 50 (1974), 521-529. MR 350741
- 85.
- J. M. Martin, A note on uncountably many disks, Pacific J. Math. 13 (1964), 1331-1333. MR 27 #6252. MR 156328
- 86.
- J. M. Martin, A rigid sphere, Fund. Math. 59 (1966), 117-121. MR 36 #7122. MR 224075
- 87.
- J. M. McPherson, Wild arcs in three-space. I: Families of Fox-Artin arcs, Pacific J. Math. 45 (1973), 585-598. MR 343276
- 88.
- E. E. Moise, Affine structures in 3-manifolds. V: The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96-114. MR 14, 72. MR 48805
- 89.
- E. E. Moise, Affine structures in 3-manifolds. VIII: Invariance of the knot-types; local tame imbedding, Ann. of Math. (2) 59 (1954), 159-170. MR 15, 889. MR 61822
- 90.
- C. D. Papakyriakopoulos, On Dehn's lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1-26. MR 19, 761. MR 90053
- 91.
- C. D. Papakyriakopoulos, On solid ton, Proc. London Math. Soc. (3) 7 (1957), 281-299. MR 19, 441. MR 87944
- 92.
- A. Schoenflies, Die Entwicklung der Lehre von den Punktmannigfaltigkeiten. Vol. II, Teubner, Leipzig, 1908.
- 93.
- C. L. Seebeck III, Tame arcs on wild cells, Proc. Amer. Math. Soc. 29 (1971), 197-201. MR 43 #6896. MR 281177
- 94.
- A. Shapiro and J. H. C. Whitehead, A proof and extension of Dehn's lemma, Bull. Amer. Math. Soc. 64 (1958), 174-178. MR 21 #2242. MR 103474
- 95.
- R. B. Sher, Tame polyhedra in wild cells and spheres, Proc. Amer. Math. Soc. 30 (1971), 169-174. MR 43 #6897. MR 281178
- 96.
- A. C. Shilepsky, Homogeneity by isotopy for simple closed curves, Duke Math. J. 40 (1973), 463-472. MR 47 #7725. MR 319179
- 97.
- A. C. Shilepsky, Homogeneity and extension properties of embeddings of S, Trans. Amer. Math. Soc. 195 (1974), 265-276. MR 341494
- 98.
- J. R. Stallings, Uncountably many wild disks, Ann. of Math. (2) 71 (1960), 185-186. MR 22 #1871. MR 111003
- 99.
- J. R. Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 12-19. MR 22 #12526. MR 121796
- 100.
- F. Waldhausen, Eine Verallgemeinerung des Schleifensatzes, Topology 6 (1967), 501-504. MR 36 #3366. MR 220300
- 101.
- J. H. C. Whitehead, On 2-spheres in 3-manifolds, Bull. Amer. Math. Soc. 64 (1958), 161-166. MR 21 #2241. MR 103473
- 102.
- G. T. Whyburn, On sequences and limiting sets, Fund. Math. 25 (1935), 408-426.
- 103.
- G. T. Whyburn, Topological characterization of the Sierpiński curve, Fund. Math. 45 (1958), 320-324, MR 20 #6077. MR 99638
- 104.
- R. L. Wilder, Partially free subsets of euclidean n-space, Michigan Math. J. 9 (1962), 97-107. MR 25 #3510. MR 140087
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(1970):
55A30, 55A35, 57A10, 57A35, 57A45, 57A50, 57C55
Retrieve articles in all Journals with MSC
(1970):
55A30, 55A35, 57A10, 57A35, 57A45, 57A50, 57C55
Additional Information:
DOI:
10.1090/S0002-9904-1975-13834-1
PII:
S 0002-9904(1975)13834-1
|