Difference between revisions of "Wild sphere"
From Encyclopedia of Mathematics
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| − | A closed manifold in Euclidean three-space | + | {{TEX|done}} |
| + | 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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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.H. Bing, "The geometric topology of | + | <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
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