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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)

     

Embeddings of $\left( {n - 1} \right)$-spheres in Euclidean $n$-space

Author(s): Robert J. Daverman
Journal: Bull. Amer. Math. Soc. 84 (1978), 377-405.
MSC (1970): Primary 57A45, 57A15, 57A35, 57A40; Secondary 57A10, 57C15, 57C30, 57C35
MathSciNet review: 0645404
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References:

[A1] J. W. Alexander, An example of a simply connected space bounding a region which is not simply connected, Proc. Nat. Acad. Sci. U. S. A. 10 (1924), 8-10.

[A2] J. W. Alexander, Remarks on a point set constructed by Antoine, Proc. Nat. Acad. Sci. U. S. A. 10 (1924), 10-12.

[A3] F. D. Ancel and J. W. Cannon, The locally flat approximation of cell-like embedding relations (to appear). MR 519353

[A4] F. D. Ancel and D. R. McMillan, Jr., Complementary 1-ULC properties for 3-spheres in 4-space, Illinois J. Math. 20 (1976), 669-680. MR 410752

[A5] J. J. Andrews and M. L. Curtis, n-space modulo an arc, Ann. of Math. (2) 75 (1962), 1-7. MR 25 #2590. MR 139153

[A6] L. Antoine, Sur l'homéomorphie de deux figures et de leur voisinages, J. Math. Pures Appl. 86 (1921), 221-325.

[A7] S. Armentrout, Monotone decompositions of E3, Topology Seminar, Wisconsin, 1965, Ann. of Math. Studies No. 60, Princeton Univ. Press, Princeton, N. J., 1966, 1-25. MR36 #5915. MR 222865

[A8] S. Armentrout, A survey of results on decompositions, Proceedings of the University of Oklahoma Topology Conference, 1972, Univ. of Oklahoma, 1972, pp. 1-12. MR 365576

[B1] Fred O. Benson, Flattening criteria for high dimensional (n - 1)-submanifolds, Ph.D. Thesis, Univ. of Utah, 1974.

[B2] Fred O. Benson, A short proof of a Kirby flattening theorem, Geometric Topology, Lecture Notes in Math., vol. 438, Springer-Verlag, Berlin and New York, 1975, pp. 6-8. MR 397741

[B3] R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 456-483. MR 19, 300. MR 87090

[B4] R. H. Bing, Conditions under which a surface in E, Fund. Math. 47 (1959), 105-139. MR 21 #5954. MR 107229

[B5] R. H. Bing, A surface is tame if its complement is 1-ULC, Trans. Amer. Math. Soc. 101 (1961), 294-305. MR 24 #A117. MR 131265

[B6] R. H. Bing, Each disk in E3 contains a tame arc, Amer. J. Math. 84 (1962), 583-590. MR 26#4331. MR 146811

[B7] R. H. Bing, Approximating surfaces from the side, Ann. of Math. (2) 77 (1963), 145-192. MR 27 #731. MR 150744

[B8] R. H. Bing, Retractions onto spheres, Amer. Math. Monthly 71 (1964), 481-484. MR 28#5435. MR 162236

[B9] R. H. Bing, Pushing a 2-sphere into its complement, Michigan Math. J. 11 (1964), 33-45. MR 28 #3408. MR 160194

[B10] R. H. Bing, Radial engulfing, Conference on Topology of Manifolds (Michigan State University, 1967), Prindle, Weber & Schmidt, Boston, 1968, pp. 1-18. MR 38 #6560. MR 238284

[B11] W. A. Blankenship, Generalization of a construction of Antoine, Ann. of Math. (2) 53 (1951), 276-291. MR 12, 730. MR 40659

[B12] W. S. Boyd, Jr., Repairing embeddings of 3-cells with monotone maps of E3, Trans. Amer. Math. Soc. 161 (1971), 123-144. MR43 #8064. MR 282352

[B13] L. E. J. Brouwer, Beweis des Jordanschen Satzes für den n-dimensionalen Räum, Math. Ann. 71 (1912), 314-319.

[B14] Morton Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. MR 22 #8470b. MR 117695

[B15] Morton Brown, Locally flat imbeddings of topological manifolds, Ann. of Math. (2) 75 (1962), 331-341. MR 24 #A3637. MR 133812

[B16] Morton Brown, Wild cells and spheres in higher dimensions, Michigan Math. J. 14 (1967), 219-224. MR 36 #4533. MR 221481

[B17] J. L. Bryant, Concerning uncountable families of n-cells in E, Michigan Math. J. 15 (1968), 477-479. MR 38 #6561. MR 238285

[B18] J. L. Bryant, An example of a wild (n - 1)-sphere in E 2-complex is tame, Proc. Amer. Math. Soc. 36 (1972), 283-288. MR 47 #7747. MR 319202

[B19] J. L. Bryant, R. D. Edwards, and C. L. Seebeck III, Approximating codimension one submanifolds with 1-LC embeddings, Notices Amer. Math. Soc. 20 (1973), A-28. Abstract #73T-G17.

[B20] J. L. Bryant and R. C. Lacher, Embeddings with mapping cylinder neighborhoods, Topology 14 (1974), 191-201. MR 394680

[B21] J. L. Bryant, R. C. Lacher, and B. J. Smith, Free spheres with mapping cylinder neighborhoods, Geometric Topology, Lecture Notes in Math., vol. 438, Springer-Verlag, Berlin and New York, 1975, pp. 58-65. MR 391105

[B22] J. L. Bryant and C. L. Seebeck III, Locally nice embeddings of polyhedra, Quart. J. Math. Oxford Ser. (2) 19 (1968), 257-274. MR 38 #2751. MR 234434

[B23] J. L. Bryant and C. L. Seebeck III, Locally nice embeddings in codimension three, Quart. J. Math. Oxford Ser. (2) 21 (1970), 265-272. MR 44 #7560. MR 290376

[B24] C. E. Burgess, Characterizations of tame surfaces in E3, Trans. Amer. Math. Soc. 114 (1965), 80-97. MR 31 #728. MR 176456

[B25] C. E. Burgess, Criteria for a 2-sphere to be tame modulo two points, Michigan Math. J. 14 (1967), 321-330. MR 35 #7314. MR 216481

[B26] C. E. Burgess, Embeddings of surfaces in Euclidean three-space, Bull. Amer. Math. Soc. 81 (1975), 795-818. MR 375319

[B27] C. E. Burgess and J. W. Cannon, Embeddings of surfaces in E3, Rocky Mountain J. Math. 1 (1971), 259-344. MR 43 #4008. MR 278277

[C1] J. W. Cannon, Characterizations of taming sets on 2-spheres, Trans. Amer. Math. Soc. 147 (1970), 289-299. MR 41 #2644. MR 257996

[C2] J. W. Cannon, *-taming sets for crumpled cubes. I. Basic properties, Trans. Amer. Math. Soc. 161 (1971), 429-440. MR 43 #8065. MR 282353

[C3] J. W. Cannon, Characterizations of tame subsets of 2-spheres in E3, Amer. J. Math. 94 (1972), 173-188. MR 45 #5980. MR 296921

[C4] J. W. Cannon, ULC properties in neighborhoods of embedded surfaces and curves in E3, Canad. J. Math. 25 (1973), 31-73. MR 47 #2589. MR 314037

[C5] J. W. Cannon, A positional characterization of the (n - 1)-dimensional Sierpiński curve in S (n ≠ 4), Fund. Math. 79 (1973), 107-112. MR 47 #7748. MR 319203

[C6] J. W. Cannon, Taming cell-like embedding relations, Geometric Topology, Lecture Notes in Math., vol. 438, Springer-Verlag, Berlin and New York, 1975, pp. 66-118. MR 391104

[C7] J. C. Cantrell, Almost locally flat embeddings of S in S, Bull. Amer. Math. Soc. 69 (1963), 716-718. MR 27 #4237. MR 154288

[C8] J. C. Cantrell and R. C. Lacher, Some local flatness criteria for low codimensional submanifolds, Quart. J. Math. Oxford Ser. (2) 21 (1970), 129-136. MR 41 #6621. MR 261608

[C9] J. C. Cantrell, T. M. Price and T. B. Rushing, A class of embeddings of S and B, Proc. Amer. Math. Soc. 29 (1971), 208-210. MR 43 #1201. MR 275445

[C10] A. V. Černavskiĭ, The k-stability of homeomorphisms and the union of cells, Dokl. Akad. Nauk. SSSR 180 (1968), 1045-1047 = Soviet Math. Dokl. 9 (1968), 729-732. MR 37 #6919. MR 231364

[C11] A. V. Černavskiĭ, Locally homotopicalty unknotted embeddings of manifolds, Dokl. Akad. Nauk SSSR 181 (1968), 290-293 = Soviet Math. Dokl. 9 (1968), 835-839. MR 38 #720. MR 232395

[C12] A. V. Černavskiĭ, Homeomorphisms of Euclidean space and topological embeddings of polyhedra in Euclidean spaces. III, Mat. Sb. 75 (117) (1968), 264-295 = Math. USSR Sb. 4 (1968), 241-266. MR 39 #6287. MR 244974

[C13] A. V. Černavskiĭ, The equivalence of local flatness and local 1-connectedness for imbeddings of (n - 1)- dimensional manifolds in n-dimensional manifolds, Mat. Sb. 91 (133) (1973), 279-286 = Math. USSR Sb. 20 (1973), 297-304. MR 48 # 12541. MR 334222

[C14] J. Cobb, Visible taming of 2-spheres in E3 (to appear).

[C15] E. H. Connell, Approximating stable homeomorphisms with piecewise linear ones, Ann. of Math. (2) 76 (1963), 326-338. MR 27 #4238. MR 154289

[C16] R. Connelly, A newproof of Brown's collaring theorem, Proc. Amer. Math. Soc. 27 (1971), 180-182. MR 42 #2490. MR 267588

[C17] M. L. Curtis and E. C. Zeeman, On the polyhedral Schoenflies theorem, Proc. Amer. Math. Soc. 11 (1960), 888-889. MR 22 #9974. MR 119208

[D1] R. J. Daverman, Slicing theorems for n-spheres in Euclidean (n + 1)-space, Trans. Amer. Math. Soc. 166 (1972), 479-489. MR 45 #4422. MR 295356

[D2] R. J. Daverman, Pushing an (n - 1)-sphere in S, Duke Math. J. 39 (1972), 719-723. MR 47 #2607. MR 314055

[D3] R. J. Daverman, On the scarcity of tame disks in certain wild cells, Fund. Math. 79 (1973), 63-77. MR 48 #5085. MR 326742

[D4] R. J. Daverman, Locally nice codimension one manifolds are locally flat, Bull. Amer. Math. Soc. 79 (1973), 410-413. MR 47 #9628. MR 321095

[D5] R. J. Daverman, Factored codimension one cells in Euclidean n-space, Pacific J. Math. 46 (1973), 37-43. MR 368015

[D6] R. J. Daverman, Approximating polyhedra in codimension one spheres embedded in S, Pacific J. Math. 51 (1974), 417-426. MR 362326

[D7] R. J. Daverman, Wild spheres that are locally flat modulo tame Cantor sets, Trans. Amer. Math. Soc. 206 (1975), 347-359. MR 375329

[D8] R. J. Daverman, On the absence of tame disks in certain wild cells, Geometric Topology, Lecture Notes in Math., vol. 438, Springer-Verlag, Berlin and New York, 1975, pp. 142-155. MR 400236

[D9] R. J. Daverman, A summary of results and problems concerning flatness of codimension one spheres in E, Geometric Topology, Lecture Notes in Math., vol. 438, Springer-Verlag, Berlin and New York, 1975, pp. 156-165. MR 407847

[D10] R. J. Daverman, Singular regular neighborhoods and local flatness in codimension one, Proc. Amer. Math. Soc. 57 (1976), 357-362. MR 420630

[D11] R. J. Daverman, Flattening a codimension one manifold by piercing with cells (to appear).

[D12] R. J. Daverman, Sewings of closed n-cell-complements (to appear).

[D13] R. J. Daverman, Every crumpled n-cube is a closed n-cell-complement (to appear). MR 488066

[D14] R. J. Daverman, Special approximations to embeddings of codimension one spheres (to appear). MR 499630

[D15] R. J. Daverman and W. T. Eaton, The cartesian product of a line with the union of two crumpled cubes, Notices Amer. Math. Soc. 15 (1968), 542. Abstract #68T-349.

[D16] R. J. Daverman and S. A. Pax, Cross sectionally simple spheres can be wild (to appear). MR 527840

[E1] W. T. Eaton, Cross sectionally simple spheres, Bull. Amer. Math. Soc. 75 (1969), 375-378. MR 39 #957. MR 239600

[E2] W. T. Eaton, Taming a surface by piercing with disks, Proc. Amer. Math. Soc. 22 (1969), 724-727. MR 39 #7579. MR 246275

[E3] W. T. Eaton, The sum of solid spheres, Michigan Math. J. 19 (1972), 193-207. MR 46 #8227. MR 309116

[E4] W. T. Eaton, A generalization of the dogbone space to E, Proc. Amer. Math. Soc. 39 (1973), 379-387. MR 48 #1238. MR 322877

[E5] R. D. Edwards, Demension theory. I, Geometric Topology, Lecture Notes in Math., vol. 438, Springer-Verlag, Berlin and New York, 1975, pp. 195-211. MR 394678

[E6] R. D. Edwards, The double suspension of a certain homology 3-sphere is S3, Notices Amer. Math. Soc. 22 (1975), A-334. Abstract #75T-G33.

[E7] R. D. Edwards, Approximating certain cell-like maps by homeomorphisms, (preprint).

[F1] 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

[H1] O. G. Harrold and C. L. Seebeck III, Locally weakly flat spaces, Trans. Amer. Math. Soc. 138 (1969), 407-414. MR 39 #954. MR 239597

[H2] J. Hempel, A surface in S3 is tame if it can be deformed into each complementary domain, Trans. Amer. Math. Soc. 111 (1964), 273-287. MR 28 #3409. MR 160195

[H3] N. Hosay, The sum of a real cube and a crumpled cube is S3, Notices Amer. Math. Soc. 10 (1963), 666. Abstract #607-17. See also errata 11 (1964), 152.

[H4] N. Hosay, A proof of the slicing theorem for 2-spheres, Bull. Amer. Math. Soc. 75 (1969), 370-374. MR 39 #956. MR 239599

[H5] W. C. Hsiang and C. T. C. Wall, On homotopy tori. II, Bull. London Math. Soc. 1 (1969), 341-342. MR 41 #2691. MR 258044

[H6] T. Hutchinson, Two point spheres are flat, Notices Amer. Math. Soc. 14 (1967), 344. Abstract #644-18.

[J1] 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

[K1] R. C. Kirby, On the set of non-locally flat points of a submanifold of codimension one, Ann. of Math. (2) 88 (1968), 281-290. MR 38 #5193. MR 236900

[K2] R. C. Kirby, The union of flat (n - 1)-balls is flat in R, Bull. Amer. Math. Soc. 74 (1968), 614-617. MR 37 #922. MR 225328

[K3] R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742-749. MR 39 #3500. MR 242166

[K4] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Ann. of Math. Studies No. 88, Princeton Univ. Press, Princeton, N.J., 1977. MR 645390

[K5] V. L. Klee, Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30-45. MR 16, 1030. MR 69388

[L1] L. L. Lininger, Some results on crumpled cubes, Trans. Amer. Math. Soc. 118 (1965), 534-549. MR 31 #2717. MR 178460

[M1] D. R. McMillan, Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327-337. MR 28 #4528. MR 161320

[M2] D. R. McMillan, Jr., An arc in M ≥ 4 (to appear).

[M3] B. Mazur, On embeddings of spheres, Bull. Amer. Math. Soc. 65 (1959), 59-65. MR 22 #8469. MR 117693

[O1] R. P. Osborne, Embedding Cantor sets in a manifold. II. An extension theorem for homeomorphisms on Cantor sets, Fund. Math. 65 (1969), 147-151. MR 40 #484. MR 247620

[P1] T. M. Price and C. L. Seebeck III, Somewhere locally flat codimension one manifolds with 1-ULC complements are locally flat, Trans. Amer. Math. Soc. 193 (1974), 111-122. MR 49 #11520. MR 346796

[R1] T. B. Rushing, Topological embeddings, Academic Press, New York, 1973. MR 50 # 1247. MR 348752

[S1] A. Schoenflies, Die Entwicklung der Lehre von den Punktmannigfaltigkeiten, vol. II, Tubner, Leipzig, 1908.

[S2] C. L. Seebeck III, Collaring an (n - 1)-manifold in an n-manifold, Trans. Amer. Math. Soc. 148 (1970), 63-68. MR 41 #2692. MR 258045

[S3] C. L. Seebeck III, Tame arcs on wild cells, Proc. Amer. Math. Soc. 29 (1971), 197-201. MR 43 #6896. MR 281177

[S4] C. L. Seebeck III, Codimension one manifolds that are locally homotopically unknotted on one side, I, II (to appear).

[S5] R. B. Sher, Tame polyhedra in wild cells and spheres, Proc. Amer. Math. Soc. 30 (1971), 169-174. MR 43 #6897. MR 281178

[S6] L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271-294. MR 45 #4431. MR 295365

[S7] L. C. Siebenmann, Topological manifolds, Proc. Internat. Congr. Math. (Nice, 1970), Vol. 2, Gauthier-Villars, Paris, 1971, pp. 133-163. MR 423356

[S8] S. Smale, Generalized Poincare's conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391-406. MR 23 #A2220. MR 137124

[S9] M. A. Štan'ko, The embedding of compacta in Euclidean space, Mat. Sb. 83 (125) (1970), 234-255 = Math. USSR Sb. 12 (1970), 234-254. MR 42 #6804. MR 271923

[S10] M. A. Štan'ko, Approximations of compacta in E, Mat. Sb. 90 (132) (1973), 625-636 = Math. USSR Sb. 19 (1973), 615-626. MR 49 #3957. MR 339194

[S11] M. A. Štan'ko, Approximations of embedded manifolds in codimension one, Mat. Sb. 94 (136) (1974), 483-494. MR 370596

[W1] Lawrence R. Weill, A new characterization of tame 2-spheres in E3, Trans. Amer. Math. Soc. 190 (1974), 243-252. MR 339188

[W2] R. L. Wilder, Topology of manifolds, Amer. Math. Soc. Colloq. Publ. no. 32, Amer. Math. Soc., Providence, R.I., 1963. MR 32 #446. MR 182958


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Additional Information:

DOI: 10.1090/S0002-9904-1978-14476-0
PII: S 0002-9904(1978)14476-0




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