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Difference between revisions of "Wild sphere"

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A closed manifold in Euclidean three-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097990/w0979901.png" /> obtained by a [[Wild imbedding|wild imbedding]] of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097990/w0979902.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097990/w0979903.png" />. Thus, a wild sphere is the sum of two discs with a common boundary, which is a [[Wild knot|wild knot]]. The first example of a wild sphere is the so-called  "horned sphere of Alexanderhorned sphere"  or Alexander sphere (Fig. a); it bounds a domain which is not homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097990/w0979904.png" /> (in the figure this is the interior of the cylinder without any interlinking handles and points forming their boundary). Fig. b shows a wild sphere in which the exterior domain alone is not homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097990/w0979905.png" />.
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A closed manifold in Euclidean three-space $E^3$ obtained by a [[Wild imbedding|wild imbedding]] of the sphere $S^2$ in $E^3$. Thus, a wild sphere is the sum of two discs with a common boundary, which is a [[Wild knot|wild knot]]. The first example of a wild sphere is the so-called  "horned sphere of Alexanderhorned sphere"  or Alexander sphere (Fig. a); it bounds a domain which is not homeomorphic to $E^3$ (in the figure this is the interior of the cylinder without any interlinking handles and points forming their boundary). Fig. b shows a wild sphere in which the exterior domain alone is not homeomorphic to $E^3$.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/w097990a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/w097990a.gif" />
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bing,  "The geometric topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097990/w0979906.png" />-manifolds" , Amer. Math. Soc.  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.E. Moise,  "Geometric topology in dimensions 2 and 3" , Springer  (1977)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bing,  "The geometric topology of $3$-manifolds" , Amer. Math. Soc.  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.E. Moise,  "Geometric topology in dimensions 2 and 3" , Springer  (1977)</TD></TR></table>

Revision as of 08:46, 12 April 2014

A closed manifold in Euclidean three-space $E^3$ obtained by a wild imbedding of the sphere $S^2$ in $E^3$. Thus, a wild sphere is the sum of two discs with a common boundary, which is a wild knot. The first example of a wild sphere is the so-called "horned sphere of Alexanderhorned sphere" or Alexander sphere (Fig. a); it bounds a domain which is not homeomorphic to $E^3$ (in the figure this is the interior of the cylinder without any interlinking handles and points forming their boundary). Fig. b shows a wild sphere in which the exterior domain alone is not homeomorphic to $E^3$.

Figure: w097990a

Figure: w097990b

References

[1] L.V. Keldysh, "Topological imbeddings in Euclidean space" Proc. Steklov Inst. Math. , 81 (1968) Trudy Mat. Inst. Akad. Nauk. SSSR , 81 (1966)


Comments

References

[a1] R.H. Bing, "The geometric topology of $3$-manifolds" , Amer. Math. Soc. (1983)
[a2] E.E. Moise, "Geometric topology in dimensions 2 and 3" , Springer (1977)
How to Cite This Entry:
Wild sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wild_sphere&oldid=16644
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article