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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ \sp{\ast} $-taming sets for crumpled cubes. III. Horizontal sections in $ 2$-spheres


Author: James W. Cannon
Journal: Trans. Amer. Math. Soc. 161 (1971), 447-456
MSC: Primary 54.78
MathSciNet review: 0282355
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Abstract: We prove that a 2-sphere S in $ {E^3}$ is tame if each horizontal section of S has at most four components. Since there are wild spheres in $ {E^3}$ whose horizontal sections have at most five components, this result is, in a sense, best possible. Much can nevertheless be said, however, even if certain sections have more than five components; and we show that the wildness of a 2-sphere S in $ {E^3}$ is severely restricted by the requirement that each of the horizontal sections of S have at most finitely many components that separate S.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0282355-0
PII: S 0002-9947(1971)0282355-0
Keywords: Taming sets, $ ^ \ast $-taming sets, crumpled cubes, slices in 2-spheres, surfaces in 3-manifolds, 2-spheres in $ {E^3}$
Article copyright: © Copyright 1971 American Mathematical Society