Isotopy

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{{authors|Askopenkov}}
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{{Authors|Askopenkov}}
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== Introduction ==
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==Definition==
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<wikitex>;
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<wikitex>;
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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.
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We work in the [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] or [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) or
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[[Wikipedia:Topological_manifold|topological]] (TOP) categories.
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If a category is omitted, then the result holds (or a definition is given) in all the three categories.
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We write `CAT homeomorphism' to mean `diffeomorphism' for CAT=DIFF, `PL homemomorphism' for CAT=PL and
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`homeomorphism' for CAT=TOP.
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Here CAT coincides with the category (of manifolds and their maps), which is omitted elsewhere in the sentence involving `CAT homeomorphism'.
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By a homeomorphism we mean a homeomorphism onto (as opposed to an embedding).
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All manifolds are assumed to be compact.
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</wikitex>
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== Ambient and non-ambient isotopy ==
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<wikitex>;
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{{beginthm|Definition|(Ambient isotopy)}}\label{dai}
{{beginthm|Definition|(Ambient isotopy)}}\label{dai}
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For manifolds $M,N$ two [[Embedding_(simple_definition)|embeddings]] $f,g:N\to M$ are called [[Wikipedia:Ambient_isotopy|ambiently isotopic]], if there exists a CAT homeomorphism $F:M\times I\to M\times I$ such that
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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
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* $F(y,0)=(y,0)$ for each $y\in M,$
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* $F(y,0)=(y,0)$ for all $y\in M,$
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* $F(f(x),1)=(g(x),1)$ for each $x\in N,$ and
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* $F(f(x),1)=(g(x),1)$ for all $x\in N,$ and
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* $F(M\times\{t\})=M\times\{t\}$ for each $t\in I.$
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* $F(M\times\{t\})=M\times\{t\}$ for all $t \in I.$
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See \cite[Figure 1.1]{Skopenkov2006}.
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[[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}-->
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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}).
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This defines an equivalence relation on the set of embeddings of $N$ into $M$ \cite{Hirsch1976} which is called `ambient isotopy'.
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{{endthm}}
{{endthm}}
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An ''ambient isotopy'' is also the above homeomorphism $F$, or, equivalently, a family of homeomorphisms $F_t:M\to M$ generated by the map $F$ in the obvious manner. The latter family can be seen as a homotopy $M\times I\to M$.
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The [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|classification of embeddings up to ambient isotopy]] is a classical problem in topology, see \cite[$\S$1]{Skopenkov2016c}.
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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}.
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<!--Ambient isotopy is a stronger equivalence relation than any of the relations non-ambient isotopy, isoposition, concordance, bordism, etc., see below. -->
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Ambient isotopy is a stronger equivalence relation than any of the relations non-ambient isotopy, isoposition, concordance, bordism, etc., see below.
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{{beginthm|Remark}}
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Some authors abbreviate ''ambient isotopy'' to just ''isotopy''. Readers should be careful to clarify the meaning of ''isotopy'' in a particular text.
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{{endthm|Remark}}
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The words ''ambient isotopy'' are often abbreviated to just ''isotopy''. One should be careful because ''isotopy'' often stands for ''non-ambient isotopy''.
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{{beginthm|Definition|(Isotopy)}}\label{di}
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<!-- or for ''homotopy in the class of embeddings''.-->
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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
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{{beginthm|Definition|(Non-ambient isotopy)}}\label{di}
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For manifolds $M,N$ two embeddings $f,g:N\to M$ are called ''non-ambient isotopic'', if there exists an embedding $F:N\times I\to M\times I$ such that
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* $F(x,0)=(f(x),0)$,
* $F(x,0)=(f(x),0)$,
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* $F(x,1)=(g(x),1)$ for each $x\in N$ and
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* $F(x,1)=(g(x),1)$ for all $x\in N$ and
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* $F(N\times\{t\})\subset M\times\{t\}$ for each $t\in I$.
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* $F(N\times\{t\})\subset M\times\{t\}$ for all $t\in I$.
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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}}
{{endthm}}
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For more delicate questions involving ''homotopy in the space of embeddings'' and non-compact manifolds see \cite{Geiges2018}.
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{{beginthm|Remark}}
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In the smooth category ''isotopy'' is also called ''diffeotopy'' by some authors.
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{{beginthm|Theorem}}\label{t:aivsi} In the smooth category, or for $m-n\ge3$ in the PL or TOP category, non-ambient isotopy implies ambient isotopy \cite[$\S$8.1]{Hirsch1976}, \cite{Hudson&Zeeman1964}, \cite{Hudson1966}, \cite{Akin1969}, \cite{Edwards1975}.
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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}.
{{endthm}}
{{endthm}}
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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.
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{{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}.
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{{endthm}}
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In the smooth category, ''non-ambient isotopy'' is also called ''diffeotopy''.
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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.
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</wikitex>
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<!--In the smooth category, ''non-ambient isotopy'' is also called ''diffeotopy''.-->
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== Isoposition and concordance ==
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<wikitex>;
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{{beginthm|Definition|(Isoposition)}}\label{dipn}
{{beginthm|Definition|(Isoposition)}}\label{dipn}
Line 62: Line 46:
For embeddings into $\Rr^m$ PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) \cite[3.22]{Rourke&Sanderson1972}.
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.
It would be interesting to know if the smooth analogue of this result holds.
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</wikitex>
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==Concordance ==
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<wikitex>;
{{beginthm|Definition|(Concordance)}}\label{dac}
{{beginthm|Definition|(Concordance)}}\label{dac}

Latest revision as of 04:40, 24 April 2019

The user responsible for this page is Askopenkov. No other user may edit this page at present.

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 I = [0,1] denotes the unit interval.

Definition 1.1 (Ambient isotopy). 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: the picture is realistic for N = S^1 and M = \R^2
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 
Tex syntax error
 (in the smooth category this is non-trivial and proven in [Hirsch1976, \S8, 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.

Definition 1.3 (Isotopy). 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 
Tex syntax error
 (in the smooth category this is non-trivial, see [Hirsch1976, \S8, 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
 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 
Tex syntax error
 into N coincides with the path components of the space of embeddings of 
Tex syntax error
 into N. For details on the space of embeddings and for information in the case of non-compact manifolds see [Geiges2018].

Theorem 1.5. In the smooth category, or for m-n\ge3 in the PL or TOP category, isotopy implies ambient isotopy [Hirsch1976, \S8.1], [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.

Definition 1.6 (Isoposition). 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

Definition 2.1 (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, \S1].

3 References

]{Skopenkov2016c}. Ambient isotopy is a stronger equivalence relation than any of the relations non-ambient isotopy, isoposition, concordance, bordism, etc., see below. The words ''ambient isotopy'' are often abbreviated to just ''isotopy''. One should be careful because ''isotopy'' often stands for ''non-ambient isotopy''. {{beginthm|Definition|(Non-ambient isotopy)}}\label{di} For manifolds $M,N$ two embeddings $f,g:N\to M$ are called ''non-ambient 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 each $x\in N$ and * $F(N\times\{t\})\subset M\times\{t\}$ for each $t\in I$. {{endthm}} For more delicate questions involving ''homotopy in the space of embeddings'' and non-compact manifolds see \cite{Geiges2018}. {{beginthm|Theorem}}\label{t:aivsi} In the smooth category, or for $m-n\ge3$ in the PL or TOP category, non-ambient isotopy implies ambient isotopy \cite[$\S.1]{Hirsch1976}, \cite{Hudson&Zeeman1964}, \cite{Hudson1966}, \cite{Akin1969}, \cite{Edwards1975}. {{endthm}} 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. In the smooth category, ''non-ambient isotopy'' is also called ''diffeotopy''. == Isoposition and concordance == ; {{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}} 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. {{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. 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 [[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[$\SI = [0,1] denotes the unit interval.

Definition 1.1 (Ambient isotopy). 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: the picture is realistic for N = S^1 and M = \R^2
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 
Tex syntax error
 (in the smooth category this is non-trivial and proven in [Hirsch1976, \S8, 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.

Definition 1.3 (Isotopy). 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 
Tex syntax error
 (in the smooth category this is non-trivial, see [Hirsch1976, \S8, 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
 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 
Tex syntax error
 into N coincides with the path components of the space of embeddings of 
Tex syntax error
 into N. For details on the space of embeddings and for information in the case of non-compact manifolds see [Geiges2018].

Theorem 1.5. In the smooth category, or for m-n\ge3 in the PL or TOP category, isotopy implies ambient isotopy [Hirsch1976, \S8.1], [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.

Definition 1.6 (Isoposition). 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

Definition 2.1 (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, \S1].

3 References

]{Skopenkov2016c}. == References == {{#RefList:}} [[Category:Definitions]] [[Category:Embeddings of manifolds]]I = [0,1] denotes the unit interval.

Definition 1.1 (Ambient isotopy). 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: the picture is realistic for N = S^1 and M = \R^2
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 
Tex syntax error
 (in the smooth category this is non-trivial and proven in [Hirsch1976, \S8, 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.

Definition 1.3 (Isotopy). 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 
Tex syntax error
 (in the smooth category this is non-trivial, see [Hirsch1976, \S8, 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
 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 
Tex syntax error
 into N coincides with the path components of the space of embeddings of 
Tex syntax error
 into N. For details on the space of embeddings and for information in the case of non-compact manifolds see [Geiges2018].

Theorem 1.5. In the smooth category, or for m-n\ge3 in the PL or TOP category, isotopy implies ambient isotopy [Hirsch1976, \S8.1], [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.

Definition 1.6 (Isoposition). 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

Definition 2.1 (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, \S1].

3 References

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