Isotopy

Latest revision as of 04:40, 24 April 2019

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 ${{Authors|Askopenkov}} ==Definition== <wikitex>; This page defines isotopies and and ambient isotopies between embeddings in either in either the [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] (DIFF), [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) or [[Wikipedia:Topological_manifold|topological]] (TOP) categories. <!--If a category is omitted, then a definition is given or a result holds all the three categories. --> 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. <!--Here CAT coincides with the category (of manifolds and their maps), which is omitted elsewhere in the sentence involving `CAT homeomorphism'. --> <!--By a homeomorphism we mean a homeomorphism onto (as opposed to an embedding).--> {{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:1-1.pdf|thumb|400px|An ambient isotopy for $M=\Rr^m$]] <!--\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, Theorem 1.9]{Hirsch1976}). <!--This equivalence relation is called `ambient isotopy'. --> {{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 $N = \R^n$ and also a summary of theorems stating when all embeddings $M \to \R^n$ are isotopic, see \cite{Skopenkov2016c}. <!--The above CAT homeomorphism $F$, or the family (=the set) of CAT homeomorphisms $F_t:M\to M$ defined by $F_t(x):=F(t,x)$, are also called an ''ambient isotopy''. --> <!--The latter family can be seen as a homotopy $M\times I\to M$.--> <!--Ambient isotopy is a stronger equivalence relation than any of the relations non-ambient isotopy, isoposition, concordance, bordism, etc., see below. --> {{beginthm|Remark}} Some authors abbreviate ''ambient isotopy'' to just ''isotopy''. Readers should be careful to clarify the meaning of ''isotopy'' in a particular text. {{endthm|Remark}} {{beginthm|Definition|(Isotopy)}}\label{di} 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 ambient 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, Theorem 1.9 and Excercise 1]{Hirsch1976}). <!--This equivalence relation is called `ambient isotopy'. --> {{endthm}} {{beginthm|Remark}} 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}} {{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.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 PL isotopic to the unknot, but is not necessarily PL ambient isotopic to the unkot. <!--In the smooth category, ''non-ambient isotopy'' is also called ''diffeotopy''.--> </wikitex> == Isoposition and concordance == <wikitex>; {{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 $M$ (in the smooth category this is non-trivial and proven in [Hirsch1976, $\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].

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 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

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.

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, $\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].

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 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

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.