Almost locally tame $2$manifolds in a $3$manifold
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 Trans. Amer. Math. Soc. 156 (1971), 5971 Request permission
Abstract:
Several conditions are given which together imply that a 2manifold M in a 3manifold is locally tame from one of its complementary domains, U, at all except possibly one point. One of these conditions is that certain arbitrarily small simple closed curves on M can be collared from U. Another condition is that there exists a certain sequence ${M_1},{M_2}, \ldots$ of 2manifolds in U converging to M with the property that each unknotted, sufficiently small simple closed curve on each ${M_i}$ is nullhomologous on ${M_i}$. Moreover, if each of these simple closed curves bounds a disk on a member of the sequence, then it is shown that M is tame from $U(M \ne {S^2})$. As a result, if U is the complementary domain of a torus in ${S^3}$ that is wild from U at just one point, then U is not homeomorphic to the complement of a tame knot in ${S^3}$.References

J. W. Alexander, On the subdivision of 3space by a polyhedron, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 68.
 R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145β158. MR 61377, DOI 10.2307/1969836
 R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 465β483. MR 87090
 R. H. Bing, A surface is tame if its complement is $1$ULC, Trans. Amer. Math. Soc. 101 (1961), 294β305. MR 131265, DOI 10.1090/S00029947196101312651
 Morton Brown, The monotone union of open $n$cells is an open $n$cell, Proc. Amer. Math. Soc. 12 (1961), 812β814. MR 126835, DOI 10.1090/S00029939196101268356
 C. E. Burgess, Characterizations of tame surfaces in $E^{3}$, Trans. Amer. Math. Soc. 114 (1965), 80β97. MR 176456, DOI 10.1090/S00029947196501764562
 C. E. Burgess, Criteria for a $2$sphere in $S^{3}$ to be tame modulo two points, Michigan Math. J. 14 (1967), 321β330. MR 216481, DOI 10.1307/mmj/1028999781
 Stewart Scott Cairns, Introductory topology, Ronald Press Co., New York, 1961. MR 0119198
 J. C. Cantrell, Almost locally polyhedral $2$spheres in $S^{3}$, Duke Math. J. 30 (1963), 249β252. MR 148042, DOI 10.1215/S0012709463030278
 R. J. Daverman, Nonhomeomorphic approximations of manifolds with surfaces of bounded genus, Duke Math. J. 37 (1970), 619β625. MR 267546, DOI 10.1215/S0012709470037762
 W. T. Eaton, Taming a surface by piercing with disks, Proc. Amer. Math. Soc. 22 (1969), 724β727. MR 246275, DOI 10.1090/S00029939196902462756
 C. H. Edwards Jr., Concentricity in $3$manifolds, Trans. Amer. Math. Soc. 113 (1964), 406β423. MR 178459, DOI 10.1090/S0002994719640178459X
 Ralph H. Fox and Emil Artin, Some wild cells and spheres in threedimensional space, Ann. of Math. (2) 49 (1948), 979β990. MR 27512, DOI 10.2307/1969408
 Ralph H. Fox, On the imbedding of polyhedra in $3$space, Ann. of Math. (2) 49 (1948), 462β470. MR 26326, DOI 10.2307/1969291
 O. G. Harrold Jr. and E. E. Moise, Almost locally polyhedral spheres, Ann. of Math. (2) 57 (1953), 575β578. MR 53504, DOI 10.2307/1969738
 Tatsuo Homma, On the existence of unknotted polygons on 2manifolds in $E^3$, Osaka Math. J. 6 (1954), 129β134. MR 63672 N. Hosay, The sum of a cube and a crumpled cube is ${S^3}$, Notices Amer. Math. Soc. 10 (1963), 666. Abstract #60717.
 Lloyd L. Lininger, Some results on crumpled cubes, Trans. Amer. Math. Soc. 118 (1965), 534β549. MR 178460, DOI 10.1090/S00029947196501784607
 D. R. McMillan Jr., Neighborhoods of surfaces in $3$manifolds, Michigan Math. J. 14 (1967), 161β170. MR 212778
 Edwin E. Moise, Affine structures in $3$manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96β114. MR 48805, DOI 10.2307/1969769
 C. D. Papakyriakopoulos, On Dehnβs lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1β26. MR 90053, DOI 10.2307/1970113 M. D. Taylor, An upper bound for the number of wild points on a 2sphere, Doctoral Dissertation, Florida State University, Tallahassee, Fla., 1962.
 Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491, DOI 10.1090/coll/032
Additional Information
 © Copyright 1971 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 156 (1971), 5971
 MSC: Primary 54.78
 DOI: https://doi.org/10.1090/S00029947197102754011
 MathSciNet review: 0275401