Isotopy

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1 Definition

This page defines isotopies and ambient isotopies between embeddings in either the smooth (DIFF), piecewise-linear (PL) or topological (TOP) categories. These notions usually appear in discussions of details, so a reader is more likely to see in the literature (including Manifold Atlas) isotopy and ambient isotopy as equivalence relations, which are also defined here. By a `CAT embedding' we mean either a `smooth embedding', a `piecewise linear' embedding or a `topological embedding', depending upon the category. By a `CAT homeomorphism' we mean a `diffeomorphism' if CAT=DIFF, a `PL homemomorphism' if CAT=PL or a`homeomorphism' if CAT=TOP. All manifolds are assumed to be compact and ${{Stub}} <wikitex>; [[High codimension embeddings: classification#Introduction|Classification of embeddings up to isotopy]] is a classical problem in topology, see \cite[$\SI = [0,1]$ denotes the unit interval.

For manifolds $M,N$ an ambient isotopy between two CAT embeddings $f,g:N\to M$ is a CAT homeomorphism $F:M\times I\to M\times I$ such that

$F(y,0)=(y,0)$ for all $y\in M,$
$F(f(x),1)=(g(x),1)$ for all $x\in N,$ and
$F(M\times\{t\})=M\times\{t\}$ for all $t \in I.$

An ambient isotopy for $M=\Rr^m$ $M=\Rr^m$ : the picture is realistic for $N = S^1$ $N = S^1$ and $M = \R^2$ $M = \R^2$

Two embeddings $f$ $f$ and $g$ $g$ are called ambient isotopic if there is an ambient isotopy between them. Ambient isotopy defines an equivalence relation on the set of CAT embeddings of $N$ $N$ into

Tex syntax error

$M$ (in the smooth category this is non-trivial and proven in [Hirsch1976, $\S$ $\S$ 8, Theorem 1.9]).

For simple examples of ambient isotopic embeddings and also embeddings which are not ambient isotopic, see [Skopenkov2016c, Remark 1.3.a]. The classification of embeddings up to ambient isotopy is a classical problem in topology. For an introduction to the case when $N = \R^m$ and also a summary of theorems stating when all embeddings $M \to \R^m$ are isotopic, see [Skopenkov2016c].

Remark 1.2. Some authors abbreviate ambient isotopy to just isotopy. Readers should be careful to clarify the meaning of isotopy in a particular text.

For manifolds $M,N$ two CAT embeddings $f,g:N\to M$ are called CAT isotopic, if there exists an embedding $F:N\times I\to M\times I$ such that

$F(x,0)=(f(x),0)$ ,
$F(x,1)=(g(x),1)$ for all $x\in N$ and
$F(N\times\{t\})\subset M\times\{t\}$ for all $t\in I$ .

Two embeddings $f$ $f$ and $g$ $g$ are called isotopic if there is an isotopy between them. Isotopy defines an equivalence relation on the set of embeddings of $N$ $N$ into

Tex syntax error

$M$ (in the smooth category this is non-trivial, see [Hirsch1976, $\S$ $\S$ 8, Theorem 1.9 and Excercise 1]).

Remark 1.4. In the smooth category isotopy is also called diffeotopy by some authors.

The set of embeddings of

Tex syntax error

$M$ into $N$ $N$ can be topologised in such a way that an isotopy is equivalent to a continuous path of embeddings. In this case the set of isotopy classes of embeddings of

Tex syntax error

$M$ into $N$ $N$ coincides with the path components of the space of embeddings of

Tex syntax error

$M$ into $N$ $N$ . For details on the space of embeddings and for information in the case of non-compact manifolds see [Geiges2018].

In the smooth category, or for $m-n\ge3$ in the PL or TOP category, isotopy implies ambient isotopy [Hirsch1976, $\S$ 8.1], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975].

For $m-n\le2$ this is not so: e.g., any knot $S^1\to\Rr^3$ is PL isotopic to the unknot, but is not necessarily PL ambient isotopic to the unkot.

For manifolds $M,N$ two embeddings $f,g:N\to M$ are called (orientation preserving) isopositioned, if there is an (orientation preserving) CAT homeomorphism $h:M\to M$ such that $h\circ f=g$ .

For embeddings into $\Rr^m$ PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22]. It would be interesting to know if the smooth analogue of this result holds.

2 Concordance

For manifolds $M,N$ two embeddings $f,g:N\to M$ are called ambiently concordant, or just concordant, if there is a homeomorphism onto $F:M\times I\to M\times I$ (which is called a concordance) such that

$F(y,0)=(y,0)$ for each $y\in M$ and
$F(f(x),1)=(g(x),1)$ for each $x\in N$ .

The definition of non-ambient concordance is analogously obtained from that of non-ambient isotopy by dropping the last condition of level-preservation. Note that in knot theory non-ambient concordance is called cobordism.

In the DIFF category or for $m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy [Lickorish1965], [Hudson1970], [Hudson&Lickorish1971]. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the Knotting Problem to the relativized Embedding Problem, see [Skopenkov2016c, $\S$ 1].

3 References

[Akin1969] E. Akin, Manifold phenomena in the theory of polyhedra, Trans. Amer. Math. Soc. 143 (1969), 413–473. MR0253329 (40 #6544) Zbl 0195.53702
[Edwards1975] R. D. Edwards, The equivalence of close piecewise linear embeddings, General Topol. Appl. 5 (1975), 147–180. MR0370603 (51 #6830) Zbl 0314.57009
[Geiges2018] H. Geiges, Isotopies vis-`a-vis level-preserving embeddings, Arch. Math. 110 (2018), 197–200. Available at the arXiv:1708.09703.
[Hirsch1976] M. W. Hirsch, Differential topology., Graduate Texts in Mathematics, No. 33. Springer-Verlag., New York-Heidelberg, 1976. MR0448362 (56 #6669) Zbl 0356.57001
[Hudson&Lickorish1971] J. F. P. Hudson and W. B. R. Lickorish, Extending piecewise linear concordances, Quart. J. Math. Oxford Ser. (2) 22 (1971), 1–12. MR0290373 (44 #7557) Zbl 0219.57011
[Hudson&Zeeman1964] J. F. P. Hudson and E. C. Zeeman, On regular neighbourhoods, Proc. London Math. Soc. (3) 14 (1964), 719–745. MR0166790 (29 #4063) Zbl 0213.25002
[Hudson1966] J. F. P. Hudson, Extending piecewise-linear isotopies, Proc. London Math. Soc. (3) 16 (1966), 651–668. MR0202147 (34 #2020) Zbl 0141.40802
[Hudson1970] J. F. P. Hudson, Concordance, isotopy, and diffeotopy, Ann. of Math. (2) 91 (1970), 425–448. MR0259920 (41 #4549) Zbl 0202.54602
[Lickorish1965] W. B. R. Lickorish, The piecewise linear unknotting of cones, Topology 4 (1965), 67–91. MR0203736 (34 #3585) Zbl 0138.19003
[Rourke&Sanderson1972] C. Rourke and B. Sanderson, Introduction to piecewise-linear topology., Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 69. Berlin-Heidelberg-New York: Springer-Verlag. VIII, 1972. MR0665919 (83g:57009) Zbl 0254.57010
[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.

]{SkopenkovC}. == Definition == ; For a space $N$ two [[Embedding|embeddings]] $f,g:N\to\Rr^m$ are said to be [[Wikipedia:Ambient_isotopy|isotopic]], if there exists a homeomorphism onto $F:\Rr^m\times I\to\Rr^m\times I$ such that * $F(y,0)=(y,0)$ for each $y\in\Rr^m,$ * $F(f(x),1)=(g(x),1)$ for each $x\in N,$ and * $F(\Rr^m\times\{t\})=\Rr^m\times\{t\}$ for each $t\in I.$ See \cite[Figure 1.1]{Skopenkov2006}. An ''isotopy'' is the above homeomorphism $F$, or, equivalently, a homotopy $\Rr^m\times I\to\Rr^m$ or a family of homeomorphisms $F_t:\Rr^m\to\Rr^m$ generated by the map $F$ in the obvious manner. Evidently, isotopy is an equivalence relation on the set of embeddings of $N$ into $\Rr^m$. This notion of isotopy is also called ''ambient'' isotopy in contrast to the ''non-ambient'' isotopy defined just below. == Other equivalence relations == ; Ambient isotopy is a stronger equivalence relation than non-ambient isotopy, isoposition, concordance, bordism, etc. Two embeddings $f,g:N\to\Rr^m$ are called ''non-ambient'' isotopic, if there exists an embedding $F:N\times I\to\Rr^m\times I$ such that * $F(x,0)=(f(x),0)$, * $F(x,1)=(g(x),1)$ for each $x\in N$ and * $F(N\times\{t\})\subset\Rr^m\times\{t\}$ for each $t\in I$. In the DIFF category or for $m-n\ge3$ in the PL or TOP category non-ambient isotopy implies ambient isotopy \cite{Hirsch1976}, \cite{Hudson&Zeeman1964}, \cite{Hudson1966}, \cite{Akin1969}, \cite{Edwards1975}. For $m-n\le2$ this is not so: e.g., any knot $S^1\to\Rr^3$ is non-ambiently PL isotopic to the trivial one, but not necessarily ambiently PL isotopic to it. Two embeddings $f,g:N\to\Rr^m$ are said to be (orientation preserving) ''isopositioned'', if there is an (orientation preserving) homeomorphism $h:\Rr^m\to\Rr^m$ such that $h\circ f=g$. For embeddings into $\Rr^m$ PL orientation preserving isoposition is equivalent to PL ambient isotopy (the Alexander-Guggenheim Theorem) \cite[3.22]{Rourke&Sanderson1972}. Two embeddings $f,g:N\to\Rr^m$ are said to be ''(ambiently) concordant'' if there is a homeomorphism (onto) $F:\Rr^m\times I\to\Rr^m\times I$ (which is called a ''concordance'') such that * $F(y,0)=(y,0)$ for each $y\in\Rr^m$ and * $F(f(x),1)=(g(x),1)$ for each $x\in N$. The definition of ''non-ambient concordance'' is analogously obtained from that of non-ambient isotopy by dropping the last condition of level-preservation. In the DIFF category or for $m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy \cite{Lickorish1965}, \cite{Hudson1970}, \cite{Hudson&Lickorish1971}. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the [[High codimension embeddings: classification#Introduction|Knotting Problem]] to the relativized [[High codimension embeddings: classification#Introduction|Embedding Problem]], see \cite[\S1]{SkopenkovC}. == References == {{#RefList:}} [[Category:Definitions]] [[Category:Embeddings of manifolds]]I = [0,1] denotes the unit interval.

For manifolds $M,N$ an ambient isotopy between two CAT embeddings $f,g:N\to M$ is a CAT homeomorphism $F:M\times I\to M\times I$ such that

$F(y,0)=(y,0)$ for all $y\in M,$
$F(f(x),1)=(g(x),1)$ for all $x\in N,$ and
$F(M\times\{t\})=M\times\{t\}$ for all $t \in I.$

An ambient isotopy for $M=\Rr^m$ $M=\Rr^m$ : the picture is realistic for $N = S^1$ $N = S^1$ and $M = \R^2$ $M = \R^2$

Two embeddings $f$ $f$ and $g$ $g$ are called ambient isotopic if there is an ambient isotopy between them. Ambient isotopy defines an equivalence relation on the set of CAT embeddings of $N$ $N$ into

Tex syntax error

$M$ (in the smooth category this is non-trivial and proven in [Hirsch1976, $\S$ $\S$ 8, Theorem 1.9]).

For simple examples of ambient isotopic embeddings and also embeddings which are not ambient isotopic, see [Skopenkov2016c, Remark 1.3.a]. The classification of embeddings up to ambient isotopy is a classical problem in topology. For an introduction to the case when $N = \R^m$ and also a summary of theorems stating when all embeddings $M \to \R^m$ are isotopic, see [Skopenkov2016c].

Remark 1.2. Some authors abbreviate ambient isotopy to just isotopy. Readers should be careful to clarify the meaning of isotopy in a particular text.

For manifolds $M,N$ two CAT embeddings $f,g:N\to M$ are called CAT isotopic, if there exists an embedding $F:N\times I\to M\times I$ such that

$F(x,0)=(f(x),0)$ ,
$F(x,1)=(g(x),1)$ for all $x\in N$ and
$F(N\times\{t\})\subset M\times\{t\}$ for all $t\in I$ .

Two embeddings $f$ $f$ and $g$ $g$ are called isotopic if there is an isotopy between them. Isotopy defines an equivalence relation on the set of embeddings of $N$ $N$ into

Tex syntax error

$M$ (in the smooth category this is non-trivial, see [Hirsch1976, $\S$ $\S$ 8, Theorem 1.9 and Excercise 1]).

Remark 1.4. In the smooth category isotopy is also called diffeotopy by some authors.

The set of embeddings of

Tex syntax error

$M$ into $N$ $N$ can be topologised in such a way that an isotopy is equivalent to a continuous path of embeddings. In this case the set of isotopy classes of embeddings of

Tex syntax error

$M$ into $N$ $N$ coincides with the path components of the space of embeddings of

Tex syntax error

$M$ into $N$ $N$ . For details on the space of embeddings and for information in the case of non-compact manifolds see [Geiges2018].

In the smooth category, or for $m-n\ge3$ in the PL or TOP category, isotopy implies ambient isotopy [Hirsch1976, $\S$ 8.1], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975].

For $m-n\le2$ this is not so: e.g., any knot $S^1\to\Rr^3$ is PL isotopic to the unknot, but is not necessarily PL ambient isotopic to the unkot.

For manifolds $M,N$ two embeddings $f,g:N\to M$ are called (orientation preserving) isopositioned, if there is an (orientation preserving) CAT homeomorphism $h:M\to M$ such that $h\circ f=g$ .

For embeddings into $\Rr^m$ PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22]. It would be interesting to know if the smooth analogue of this result holds.

2 Concordance

For manifolds $M,N$ two embeddings $f,g:N\to M$ are called ambiently concordant, or just concordant, if there is a homeomorphism onto $F:M\times I\to M\times I$ (which is called a concordance) such that

$F(y,0)=(y,0)$ for each $y\in M$ and
$F(f(x),1)=(g(x),1)$ for each $x\in N$ .

The definition of non-ambient concordance is analogously obtained from that of non-ambient isotopy by dropping the last condition of level-preservation. Note that in knot theory non-ambient concordance is called cobordism.

In the DIFF category or for $m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy [Lickorish1965], [Hudson1970], [Hudson&Lickorish1971]. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the Knotting Problem to the relativized Embedding Problem, see [Skopenkov2016c, $\S$ 1].

3 References

[Akin1969] E. Akin, Manifold phenomena in the theory of polyhedra, Trans. Amer. Math. Soc. 143 (1969), 413–473. MR0253329 (40 #6544) Zbl 0195.53702
[Edwards1975] R. D. Edwards, The equivalence of close piecewise linear embeddings, General Topol. Appl. 5 (1975), 147–180. MR0370603 (51 #6830) Zbl 0314.57009
[Geiges2018] H. Geiges, Isotopies vis-`a-vis level-preserving embeddings, Arch. Math. 110 (2018), 197–200. Available at the arXiv:1708.09703.
[Hirsch1976] M. W. Hirsch, Differential topology., Graduate Texts in Mathematics, No. 33. Springer-Verlag., New York-Heidelberg, 1976. MR0448362 (56 #6669) Zbl 0356.57001
[Hudson&Lickorish1971] J. F. P. Hudson and W. B. R. Lickorish, Extending piecewise linear concordances, Quart. J. Math. Oxford Ser. (2) 22 (1971), 1–12. MR0290373 (44 #7557) Zbl 0219.57011
[Hudson&Zeeman1964] J. F. P. Hudson and E. C. Zeeman, On regular neighbourhoods, Proc. London Math. Soc. (3) 14 (1964), 719–745. MR0166790 (29 #4063) Zbl 0213.25002
[Hudson1966] J. F. P. Hudson, Extending piecewise-linear isotopies, Proc. London Math. Soc. (3) 16 (1966), 651–668. MR0202147 (34 #2020) Zbl 0141.40802
[Hudson1970] J. F. P. Hudson, Concordance, isotopy, and diffeotopy, Ann. of Math. (2) 91 (1970), 425–448. MR0259920 (41 #4549) Zbl 0202.54602
[Lickorish1965] W. B. R. Lickorish, The piecewise linear unknotting of cones, Topology 4 (1965), 67–91. MR0203736 (34 #3585) Zbl 0138.19003
[Rourke&Sanderson1972] C. Rourke and B. Sanderson, Introduction to piecewise-linear topology., Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 69. Berlin-Heidelberg-New York: Springer-Verlag. VIII, 1972. MR0665919 (83g:57009) Zbl 0254.57010
[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.

@@ Line 1: / Line 1: @@
-{{Stub}}
+{{Authors|Askopenkov}}
-<wikitex>;
+==Definition==
-[[High codimension embeddings: classification#Introduction|Classification of embeddings up to isotopy]] is a classical problem in topology, see \cite[$\S$1]{SkopenkovC}.
+<wikitex>;
-</wikitex>
+This page defines isotopies and ambient isotopies between embeddings in either the [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] (DIFF), [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) or [[Wikipedia:Topological_manifold|topological]] (TOP) categories. These notions usually appear in discussions of details, so a reader is more likely to see in the literature (including Manifold Atlas) isotopy and ambient isotopy as equivalence relations, which are also defined here. By a `CAT embedding' we mean either a `smooth embedding', a `piecewise linear' embedding or a `topological embedding', depending upon the category.  By a `CAT homeomorphism' we mean a `diffeomorphism' if CAT=DIFF, a `PL homemomorphism' if CAT=PL or a`homeomorphism' if CAT=TOP.  All manifolds are assumed to be compact and $I = [0,1]$ denotes the unit interval.
+{{beginthm|Definition|(Ambient isotopy)}}\label{dai}
+For manifolds $M,N$ an [[Wikipedia:Ambient_isotopy|ambient isotopy]] between two CAT [[Embedding_(simple_definition)|embeddings]] $f,g:N\to M$ is a CAT homeomorphism $F:M\times I\to M\times I$ such that
+* $F(y,0)=(y,0)$ for all $y\in M,$
+* $F(f(x),1)=(g(x),1)$ for all $x\in N,$ and
+* $F(M\times\{t\})=M\times\{t\}$ for all $t \in I.$
+[[Image:AmbientIsotopy.jpg|thumb|400px|An ambient isotopy for $M=\Rr^m$: the picture is realistic for $N = S^1$ and $M = \R^2$]] <!--\cite[Figure 1.1]{Skopenkov2006}-->
+Two embeddings $f$ and $g$ are called ambient isotopic if there is an ambient isotopy between them.  Ambient isotopy defines an equivalence relation on the set of CAT embeddings of $N$ into $M$ (in the smooth category this is non-trivial and proven in \cite[$\S$8, Theorem 1.9]{Hirsch1976}).
+{{endthm}}
+For simple examples of ambient isotopic embeddings and also embeddings which are not ambient isotopic, see  \cite[Remark 1.3.a]{Skopenkov2016c}.  The [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|classification of embeddings up to ambient isotopy]] is a classical problem in topology.  For an introduction to the case when $N = \R^m$ and also a summary of theorems stating when all embeddings $M \to \R^m$ are isotopic, see \cite{Skopenkov2016c}.
+<!--Ambient isotopy is a stronger equivalence relation than any of the relations non-ambient isotopy, isoposition, concordance, bordism, etc., see below. -->
-== Definition  ==
+{{beginthm|Remark}}
-<wikitex>;
+Some authors abbreviate ''ambient isotopy'' to just ''isotopy''.  Readers should be careful to clarify the meaning of ''isotopy'' in a particular text.
-For a space $N$ two [[Embedding|embeddings]] $f,g:N\to\Rr^m$ are said to be [[Wikipedia:Ambient_isotopy|isotopic]], if there exists a homeomorphism onto $F:\Rr^m\times I\to\Rr^m\times I$ such that
+{{endthm|Remark}}
-* $F(y,0)=(y,0)$ for each $y\in\Rr^m,$
-* $F(f(x),1)=(g(x),1)$ for each $x\in N,$ and
-* $F(\Rr^m\times\{t\})=\Rr^m\times\{t\}$ for each $t\in I.$
-See \cite[Figure 1.1]{Skopenkov2006}.
+{{beginthm|Definition|(Isotopy)}}\label{di}
-An ''isotopy'' is the above homeomorphism $F$, or, equivalently, a homotopy $\Rr^m\times I\to\Rr^m$ or a family of homeomorphisms $F_t:\Rr^m\to\Rr^m$ generated by the map $F$ in the obvious manner.
+For manifolds $M,N$ two CAT embeddings $f,g:N\to M$ are called ''CAT isotopic'', if there exists an embedding $F:N\times I\to M\times I$ such that
+* $F(x,0)=(f(x),0)$,
+* $F(x,1)=(g(x),1)$ for all $x\in N$ and
+* $F(N\times\{t\})\subset M\times\{t\}$ for all $t\in I$.
+Two embeddings $f$ and $g$ are called isotopic if there is an isotopy between them.  Isotopy defines an equivalence relation on the set of embeddings of $N$ into $M$ (in the smooth category this is non-trivial, see \cite[$\S$8, Theorem 1.9 and Excercise 1]{Hirsch1976}).
+{{endthm}}
-Evidently, isotopy is an equivalence relation on the set of embeddings of $N$ into $\Rr^m$.
+{{beginthm|Remark}}
+In the smooth category ''isotopy'' is also called ''diffeotopy'' by some authors.
-This notion of isotopy is also called ''ambient'' isotopy in contrast to the ''non-ambient'' isotopy defined just below.
+The set of embeddings of $M$ into $N$ can be topologised in such a way that an isotopy is equivalent to a continuous path of embeddings.  In this case the set of isotopy classes of embeddings of $M$ into $N$ coincides with the path components of the space of embeddings of $M$ into $N$.  For details on the space of embeddings and for information in the case of non-compact manifolds see \cite{Geiges2018}.
-</wikitex>
+{{endthm}}
-== Other equivalence relations  ==
+{{beginthm|Theorem}}\label{t:aivsi} In the smooth category, or for $m-n\ge3$ in the PL or TOP category, isotopy implies ambient isotopy \cite[$\S$8.1]{Hirsch1976}, \cite{Hudson&Zeeman1964}, \cite{Hudson1966}, \cite{Akin1969}, \cite{Edwards1975}.
-<wikitex>;
+{{endthm}}
-Ambient isotopy is a stronger equivalence relation than non-ambient isotopy, isoposition, concordance, bordism, etc.
-Two embeddings $f,g:N\to\Rr^m$ are called ''non-ambient'' isotopic, if there exists an embedding $F:N\times I\to\Rr^m\times I$ such that
+For $m-n\le2$ this is not so: e.g., any knot $S^1\to\Rr^3$ is PL isotopic to the unknot, but is not necessarily PL ambient isotopic to the unkot.
-* $F(x,0)=(f(x),0)$,
+<!--In the smooth category, ''non-ambient isotopy'' is also called ''diffeotopy''.-->
-* $F(x,1)=(g(x),1)$ for each $x\in N$ and
-* $F(N\times\{t\})\subset\Rr^m\times\{t\}$ for each $t\in I$.
-In the DIFF category or for $m-n\ge3$ in the PL or TOP category non-ambient isotopy implies ambient isotopy \cite{Hirsch1976}, \cite{Hudson&Zeeman1964}, \cite{Hudson1966}, \cite{Akin1969}, \cite{Edwards1975}. For $m-n\le2$ this is not so: e.g., any knot $S^1\to\Rr^3$ is non-ambiently PL isotopic to the trivial one, but not necessarily ambiently PL isotopic to it.
+{{beginthm|Definition|(Isoposition)}}\label{dipn}
+For manifolds $M,N$ two embeddings $f,g:N\to M$ are called (orientation preserving) ''isopositioned'', if there is an (orientation preserving) CAT homeomorphism $h:M\to M$ such that $h\circ f=g$.
+{{endthm}}
-Two embeddings $f,g:N\to\Rr^m$ are said to be (orientation preserving) ''isopositioned'', if there is an (orientation preserving) homeomorphism $h:\Rr^m\to\Rr^m$ such that $h\circ f=g$.
+For embeddings into $\Rr^m$ PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) \cite[3.22]{Rourke&Sanderson1972}.
+It would be interesting to know if the smooth analogue of this result holds.
+</wikitex>
-For embeddings into $\Rr^m$ PL orientation preserving isoposition is equivalent to PL ambient isotopy (the Alexander-Guggenheim Theorem) \cite[3.22]{Rourke&Sanderson1972}.
+==Concordance ==
+<wikitex>;
-Two embeddings $f,g:N\to\Rr^m$ are said to be ''(ambiently) concordant'' if there is a homeomorphism (onto) $F:\Rr^m\times I\to\Rr^m\times I$ (which is called a ''concordance'') such that
-* $F(y,0)=(y,0)$ for each $y\in\Rr^m$ and
+{{beginthm|Definition|(Concordance)}}\label{dac}
+For manifolds $M,N$ two embeddings $f,g:N\to M$ are called ''ambiently concordant'', or just ''concordant'', if there is a homeomorphism onto $F:M\times I\to M\times I$ (which is called a ''concordance'') such that
+* $F(y,0)=(y,0)$ for each $y\in M$ and
 * $F(f(x),1)=(g(x),1)$ for each $x\in N$.
+{{endthm}}
-The definition of ''non-ambient concordance'' is analogously obtained from that of non-ambient isotopy by dropping the last condition of level-preservation.
+The definition of ''non-ambient concordance'' is analogously obtained from that of non-ambient isotopy by dropping the last condition of level-preservation. Note that in knot theory ''non-ambient concordance'' is called ''cobordism''.
-In the DIFF category or for $m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy \cite{Lickorish1965}, \cite{Hudson1970}, \cite{Hudson&Lickorish1971}. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the [[High codimension embeddings: classification#Introduction|Knotting Problem]] to the relativized [[High codimension embeddings: classification#Introduction|Embedding Problem]], see \cite[\S1]{SkopenkovC}.
+In the DIFF category or for $m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy \cite{Lickorish1965}, \cite{Hudson1970}, \cite{Hudson&Lickorish1971}.
+(This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Knotting Problem]] to the relativized [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Embedding Problem]], see \cite[$\S$1]{Skopenkov2016c}.
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Isotopy

Latest revision as of 04:40, 24 April 2019

1 Definition

2 Concordance

3 References

2 Concordance

3 References

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