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Difference between revisions of "Wild imbedding"

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''of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w0979701.png" /> in a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w0979702.png" />''
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''of a topological space $X$ in a [[Topological space|topological space]] $Y$''
  
An imbedding which is topologically non-equivalent to an imbedding from a certain class of chosen imbeddings known as tame or nice imbeddings. The cases listed below are the most useful; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w0979703.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w0979704.png" /> is taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w0979705.png" />.
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An imbedding which is topologically non-equivalent to an imbedding from a certain class of chosen imbeddings known as tame or nice imbeddings. The cases listed below are the most useful; the $n$-dimensional Euclidean space $\mathbf R^n$ is taken as $Y$.
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w0979706.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w0979707.png" />-dimensional topological manifold (cf. [[Topology of manifolds|Topology of manifolds]]). A topological imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w0979708.png" /> (cf. [[Topology of imbeddings|Topology of imbeddings]]) is called wild if there does not exist a homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w0979709.png" /> onto itself which would convert <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797010.png" /> into a locally flat submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797011.png" />.
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1) Let $M$ be a $k$-dimensional topological manifold (cf. [[Topology of manifolds|Topology of manifolds]]). A topological imbedding $g\colon M\to\mathbf R^n$ (cf. [[Topology of imbeddings|Topology of imbeddings]]) is called wild if there does not exist a homeomorphism of $\mathbf R^n$ onto itself which would convert $g(M)$ into a locally flat submanifold of $\mathbf R^n$.
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797012.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797013.png" />-dimensional [[Polyhedron|polyhedron]]. A topological imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797014.png" /> is called wild if there does not exist a homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797015.png" /> onto itself which would convert <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797016.png" /> into a polyhedron (i.e. into a body having a certain triangulation) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797017.png" />.
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2) Let $P$ be a $k$-dimensional [[Polyhedron|polyhedron]]. A topological imbedding $g\colon P\to\mathbf R^n$ is called wild if there does not exist a homeomorphism of $\mathbf R^n$ onto itself which would convert $g(P)$ into a polyhedron (i.e. into a body having a certain triangulation) in $\mathbf R^n$.
  
3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797018.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797019.png" />-dimensional [[Locally compact space|locally compact space]]. A topological imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797020.png" /> is called wild if there does not exist a homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797021.png" /> onto itself which would convert <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797022.png" /> into a subset of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797023.png" />-dimensional Menger compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797024.png" />.
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3) Let $K$ be a $k$-dimensional [[Locally compact space|locally compact space]]. A topological imbedding $g\colon K\to\mathbf R^n$ is called wild if there does not exist a homeomorphism of $\mathbf R^n$ onto itself which would convert $g(K)$ into a subset of the $k$-dimensional Menger compactum $M_n^k$.
  
If the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797025.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797026.png" />, then the properties introduced in all three cases are characterized by the following locally homotopic property: An imbedding is wild if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797027.png" /> does not satisfy the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797028.png" /> (cf. [[Topology of imbeddings|Topology of imbeddings]]). The situation is much more complicated for the codimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797030.png" />: The problem has been solved for manifolds of codimension 1 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797031.png" />, but has not been fully solved for imbeddings of codimension 2 both for manifolds and for polyhedra. All that has been said is also meaningful if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797032.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097970/w09797033.png" />-dimensional manifold — topological or piecewise linear.
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If the dimension $k\leq n-3$ and if $n\geq5$, then the properties introduced in all three cases are characterized by the following locally homotopic property: An imbedding is wild if and only if $g(X)$ does not satisfy the property $1-ULC$ (cf. [[Topology of imbeddings|Topology of imbeddings]]). The situation is much more complicated for the codimensions $n-k=1$ and $2$: The problem has been solved for manifolds of codimension 1 for $n\geq6$, but has not been fully solved for imbeddings of codimension 2 both for manifolds and for polyhedra. All that has been said is also meaningful if $Y$ is an $n$-dimensional manifold — topological or piecewise linear.

Latest revision as of 08:53, 12 April 2014

of a topological space $X$ in a topological space $Y$

An imbedding which is topologically non-equivalent to an imbedding from a certain class of chosen imbeddings known as tame or nice imbeddings. The cases listed below are the most useful; the $n$-dimensional Euclidean space $\mathbf R^n$ is taken as $Y$.

1) Let $M$ be a $k$-dimensional topological manifold (cf. Topology of manifolds). A topological imbedding $g\colon M\to\mathbf R^n$ (cf. Topology of imbeddings) is called wild if there does not exist a homeomorphism of $\mathbf R^n$ onto itself which would convert $g(M)$ into a locally flat submanifold of $\mathbf R^n$.

2) Let $P$ be a $k$-dimensional polyhedron. A topological imbedding $g\colon P\to\mathbf R^n$ is called wild if there does not exist a homeomorphism of $\mathbf R^n$ onto itself which would convert $g(P)$ into a polyhedron (i.e. into a body having a certain triangulation) in $\mathbf R^n$.

3) Let $K$ be a $k$-dimensional locally compact space. A topological imbedding $g\colon K\to\mathbf R^n$ is called wild if there does not exist a homeomorphism of $\mathbf R^n$ onto itself which would convert $g(K)$ into a subset of the $k$-dimensional Menger compactum $M_n^k$.

If the dimension $k\leq n-3$ and if $n\geq5$, then the properties introduced in all three cases are characterized by the following locally homotopic property: An imbedding is wild if and only if $g(X)$ does not satisfy the property $1-ULC$ (cf. Topology of imbeddings). The situation is much more complicated for the codimensions $n-k=1$ and $2$: The problem has been solved for manifolds of codimension 1 for $n\geq6$, but has not been fully solved for imbeddings of codimension 2 both for manifolds and for polyhedra. All that has been said is also meaningful if $Y$ is an $n$-dimensional manifold — topological or piecewise linear.

How to Cite This Entry:
Wild imbedding. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wild_imbedding&oldid=18211
This article was adapted from an original article by M.A. Shtan'ko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article