Isotopy

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[[High codimension embeddings: classification#Introduction|Classification of embeddings up to isotopy]] is a classical problem in topology.
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{{Authors|Askopenkov}}
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==Definition==
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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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{{beginthm|Definition|(Ambient isotopy)}}\label{dai}
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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 all $y\in M,$
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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 all $t \in I.$
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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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{{endthm}}
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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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== Definition ==
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{{beginthm|Remark}}
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<wikitex>;
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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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Two embeddings $f,g:N\to\Rr^m$ are said to be isotopic (see \cite{Skopenkov2006}, Figure 1.1), if there exists a homeomorphism onto (an isotopy) $F:\Rr^m\times I\to\Rr^m\times I$ such that
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{{endthm|Remark}}
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* $F(y,0)=(y,0)$ for each $y\in\Rr^m,$
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* $F(f(x),1)=(g(x),1)$ for each $x\in N,$ and
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* $F(\Rr^m\times\{t\})=\Rr^m\times\{t\}$ for each $t\in I.$
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An isotopy is also 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.
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{{beginthm|Definition|(Isotopy)}}\label{di}
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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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* $F(x,0)=(f(x),0)$,
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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 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}).
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{{endthm}}
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Evidently, isotopy is an equivalence relation on the set of embeddings of $N$ into $\Rr^m$.
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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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This notion of isotopy is also called ''ambient'' isotopy in contrast to the ''non-ambient'' isotopy defined just below.
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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}.
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</wikitex>
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{{endthm}}
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== Other equivalence relations ==
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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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<wikitex>;
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{{endthm}}
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Ambient isotopy is a stronger equivalence relation than non-ambient isotopy, isoposition, concordance, bordism, etc.
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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
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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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* $F(x,0)=(f(x),0)$,
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<!--In the smooth category, ''non-ambient isotopy'' is also called ''diffeotopy''.-->
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* $F(x,1)=(g(x),1)$ for each $x\in N$ and
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* $F(N\times\{t\})\subset\Rr^m\times\{t\}$ for each $t\in I$.
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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}, \S7. 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|Definition|(Isoposition)}}\label{dipn}
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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 $h\circ f=g$.
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{{endthm}}
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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 $h\circ f=g$.
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For embeddings into $\Rr^m$ PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) \cite[3.22]{Rourke&Sanderson1972}.
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It would be interesting to know if the smooth analogue of this result holds.
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</wikitex>
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For embeddings into $\Rr^m$ PL orientation preserving isoposition is equivalent to PL ambient isotopy (the Alexander-Guggenheim Theorem) \cite{Rourke&Sanderson1972}, 3.22.
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==Concordance ==
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<wikitex>;
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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
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* $F(y,0)=(y,0)$ for each $y\in\Rr^m$ and
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{{beginthm|Definition|(Concordance)}}\label{dac}
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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
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* $F(y,0)=(y,0)$ for each $y\in M$ and
* $F(f(x),1)=(g(x),1)$ for each $x\in N$.
* $F(f(x),1)=(g(x),1)$ for each $x\in N$.
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{{endthm}}
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The definition of ''non-ambient concordance'' is analogously obtained from that of non-ambient isotopy by dropping the last condition of level-preservation.
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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''.
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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 Embedding Problem.
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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}.
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(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}.
</wikitex>
</wikitex>
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[[Category:Theory]]
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[[Category:Definitions]]
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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 M (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 M (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 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 [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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