Embedding phenomena based upon decomposition theory: locally spherical but wild codimension one spheres
Author:
Robert J. Daverman
Journal:
Proc. Amer. Math. Soc. 90 (1984), 139144
MSC:
Primary 57N50; Secondary 54B15, 57M30, 57N15, 57N45
MathSciNet review:
722432
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Abstract: For we describe an sphere wildly embedded in the sphere yet every point of has arbitrarily small neighborhoods bounded by flat spheres, each intersecting in an sphere. Not only do these examples for large run counter to what can occur when , they also illustrate the sharpness of highdimensional taming theorems developed by Cannon and Harrold and Seebeck. Furthermore, despite their wildness, they have mapping cylinder neighborhoods, which both run counter to what is possible when and also partially illustrate the sharpness of another highdimensional taming theorem due to Bryant and Lacher.
 [1]
Edward
G. Begle, The Vietoris mapping theorem for bicompact spaces,
Ann. of Math. (2) 51 (1950), 534–543. MR 0035015
(11,677b)
 [2]
J.
L. Bryant and R.
C. Lacher, Embeddings with mapping cylinder neighborhoods,
Topology 14 (1975), 191–201. MR 0394680
(52 #15479)
 [3]
C.
E. Burgess, Characterizations of tame surfaces in
𝐸³, Trans. Amer. Math. Soc. 114 (1965), 80–97.
MR
0176456 (31 #728), http://dx.doi.org/10.1090/S00029947196501764562
 [4]
J.
W. Cannon, Characterization of taming sets on
2spheres, Trans. Amer. Math. Soc. 147 (1970), 289–299. MR 0257996
(41 #2644), http://dx.doi.org/10.1090/S00029947197002579966
 [5]
J.
W. Cannon, 𝑈𝐿𝐶 properties in neighbourhoods
of embedded surfaces and curves in 𝐸³, Canad. J. Math.
25 (1973), 31–73. MR 0314037
(47 #2589)
 [6]
J.
W. Cannon, Shrinking celllike decompositions of manifolds.
Codimension three, Ann. of Math. (2) 110 (1979),
no. 1, 83–112. MR 541330
(80j:57013), http://dx.doi.org/10.2307/1971245
 [7]
A.
V. Černavskiĭ, The identity of local flatness and
local simple connectedness for imbeddings of (𝑛1)dimensional into
𝑛dimensional manifolds when 𝑛>4, Mat. Sb. (N.S.)
91(133) (1973), 279–286, 288 (Russian). MR 0334222
(48 #12541)
 [8]
Robert
J. Daverman, Locally nice codimension one manifolds
are locally flat, Bull. Amer. Math. Soc. 79 (1973), 410–413.
MR
0321095 (47 #9628), http://dx.doi.org/10.1090/S000299041973131908
 [9]
Robert
J. Daverman, Embedding phenomena based upon
decomposition theory: wild Cantor sets satisfying strong homogeneity
properties, Proc. Amer. Math. Soc.
75 (1979), no. 1,
177–182. MR
529237 (80k:57031), http://dx.doi.org/10.1090/S00029939197905292377
 [10]
W.
T. Eaton, A note about locally spherical spheres, Canad. J.
Math. 21 (1969), 1001–1003. MR 0244969
(39 #6282)
 [11]
Robert
D. Edwards, The topology of manifolds and celllike maps,
Proceedings of the International Congress of Mathematicians (Helsinki,
1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 111–127. MR 562601
(81g:57010)
 [12]
O.
G. Harrold Jr., Locally peripherally unknotted surfaces in
𝐸³, Ann. of Math. (2) 69 (1959),
276–290. MR 0105660
(21 #4399a)
 [13]
O.
G. Harrold and C.
L. Seebeck, Locally weakly flat spaces,
Trans. Amer. Math. Soc. 138 (1969), 407–414. MR 0239597
(39 #954), http://dx.doi.org/10.1090/S00029947196902395970
 [14]
V.
L. Klee Jr., Some topological properties of convex
sets, Trans. Amer. Math. Soc. 78 (1955), 30–45. MR 0069388
(16,1030c), http://dx.doi.org/10.1090/S00029947195500693885
 [15]
Chris
Lacher and Alden
Wright, Mapping cylinders and 4manifolds, Topology of
Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham,
Chicago, Ill., 1970, pp. 424–427. MR 0271951
(42 #6832)
 [16]
L.
D. Loveland, Tame surfaces and tame subsets of
spheres in 𝐸³, Trans. Amer. Math.
Soc. 123 (1966),
355–368. MR 0199850
(33 #7990), http://dx.doi.org/10.1090/S00029947196601998503
 [17]
M.
H. A. Newman, The engulfing theorem for topological manifolds,
Ann. of Math. (2) 84 (1966), 555–571. MR 0203708
(34 #3557)
 [18]
Victor
Nicholson, Mapping cylinder
neighborhoods, Trans. Amer. Math. Soc. 143 (1969), 259–268.
MR
0248788 (40 #2038), http://dx.doi.org/10.1090/S00029947196902487884
 [19]
T.
M. Price and C.
L. Seebeck III, Somewhere locally flat codimension one
manifolds with 1𝑈𝐿𝐶\ complements are locally
flat, Trans. Amer. Math. Soc. 193 (1974), 111–122. MR 0346796
(49 #11520), http://dx.doi.org/10.1090/S00029947197403467968
 [20]
Edwin
H. Spanier, Algebraic topology, McGrawHill Book Co., New
YorkToronto, Ont.London, 1966. MR 0210112
(35 #1007)
 [1]
 E. G. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. of Math. (2) 51 (1950), 534543. MR 0035015 (11:677b)
 [2]
 J. L. Bryant and R. C. Lacher, Embeddings with mapping cylinder neighborhoods, Topology 14 (1974), 191201. MR 0394680 (52:15479)
 [3]
 C. E. Burgess, Characterizations of tame surfaces in , Trans. Amer. Math. Soc. 114 (1965), 8097. MR 0176456 (31:728)
 [4]
 J. W. Cannon, Characterizations of taming sets on spheres, Trans. Amer. Math. Soc. 147 (1970), 289299. MR 0257996 (41:2644)
 [5]
 , ULC properties in neighborhoods of embedded surfaces and curves in , Canad. J. Math. 25 (1973), 3137. MR 0314037 (47:2589)
 [6]
 , Shrinking celllike decompositions of manifolds. Codimension three, Ann. of Math. (2) 110 (1979), 83112. MR 541330 (80j:57013)
 [7]
 A. V. Černavskiĭ, The equivalence of local flatness and local connectedness for dimensional manifolds in dimensional manifolds, Mat. Sb. 91 (133) (1973), 279286 = Math. USSR Sb. 20 (1973), 297304. MR 0334222 (48:12541)
 [8]
 R. J. Daverman, Locally nice codimension one manifolds are locally flat, Bull. Amer. Math. Soc. 79 (1973), 410413. MR 0321095 (47:9628)
 [9]
 , Embedding phenomena based upon decomposition theory: wild Cantor sets satisfying strong homogeneity properties, Proc. Amer. Math. Soc. 75 (1979), 177182. MR 529237 (80k:57031)
 [10]
 W. T. Eaton, A note about locally spherical spheres, Canad. J. Math. 21 (1969), 10011003. MR 0244969 (39:6282)
 [11]
 R. D. Edwards, The topology of manifolds and celllike maps, (Proc. Internat. Congr. of Mathematicians, Helsinki, 1978), O. Lehto, ed., Academia Scientarium Fennica, Helsinki, 1980, pp. 111127. MR 562601 (81g:57010)
 [12]
 O. G. Harrold, Locally peripherally unknotted surfaces in , Ann. of Math. (2) 69 (1959), 276290. MR 0105660 (21:4399a)
 [13]
 O. G. Harrold and C. L. Seebeck III, Locally weakly flat surfaces, Trans. Amer. Math. Soc. 138 (1969), 407414. MR 0239597 (39:954)
 [14]
 V. L. Klee, Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 3045. MR 0069388 (16:1030c)
 [15]
 C. Lacher and A. Wright, Mapping cylinders and manifolds, Topology of Manifolds (J. C. Cantrell and C. H. Edwards, eds.), Markham, Chicago, 1970, pp. 424427. MR 0271951 (42:6832)
 [16]
 L. D. Loveland, Tame surfaces and tame subsets of spheres in , Trans. Amer. Math. Soc. 123 (1966), 355368. MR 0199850 (33:7990)
 [17]
 M. H. A. Newman, The engulfing theorem for topological manifolds, Ann. of Math. (2) 84 (1966), 555571. MR 0203708 (34:3557)
 [18]
 V. Nicholson, Mapping cylinder neighborhoods, Trans. Amer. Math. Soc. 143 (1969), 259268. MR 0248788 (40:2038)
 [19]
 T. M. Price and C. L. Seebeck III, Somewhere locally flat codimension one manifolds are flat, Trans. Amer. Math. Soc. 193 (1974), 111122. MR 0346796 (49:11520)
 [20]
 E. H. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 0210112 (35:1007)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198407224325
PII:
S 00029939(1984)07224325
Keywords:
Wild embedding,
locally flat,
codimension one sphere,
locally spherical,
locally weakly flat,
homology cell,
upper semicontinuous decomposition
Article copyright:
© Copyright 1984
American Mathematical Society
