Isotopy

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1 Introduction

We work in the smooth or piecewise-linear (PL) or topological (TOP) category. If a category is omitted, then the result holds (or a definition is given) in all the three categories.

All manifolds are tacitly assumed to be compact.

2 Ambient and non-ambient isotopy

Definition 2.1 (Ambient isotopy). For manifolds $M,N$${{Stub}} == Introduction == We work in the [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] or [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) or [[Wikipedia:Topological_manifold|topological]] (TOP) category. If a category is omitted, then the result holds (or a definition is given) in all the three categories. All manifolds are tacitly assumed to be compact. == Ambient and non-ambient isotopy == <wikitex>; {{beginthm|Definition|(Ambient isotopy)}}\label{dai} For manifolds $M,N$ two [[Embedding_(simple_definition)|embeddings]] $f,g:N\to M$ are called [[Wikipedia:Ambient_isotopy|ambiently isotopic]], if there exists a homeomorphism onto $F:M\times I\to M\times I$ such that * $F(y,0)=(y,0)$ for each $y\in M,$ * $F(f(x),1)=(g(x),1)$ for each $x\in N,$ and * $F(M\times\{t\})=M\times\{t\}$ for each $t\in I.$ {{endthm}} See \cite[Figure 1.1]{Skopenkov2006}. An ''ambient isotopy'' is the above homeomorphism $F$, or, equivalently, a family of homeomorphisms $F_t:M\to M$ generated by the map $F$ in the obvious manner. The latter family can be seen as a homotopy $M\times I\to M$. Evidently, ambient isotopy is an equivalence relation on the set of embeddings of $N$ into $M$. [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Classification of embeddings up to ambient isotopy]] is a classical problem in topology, see \cite[$\SM,N$ two embeddings $f,g:N\to M$$f,g:N\to M$ are called ambiently isotopic, if there exists a homeomorphism onto $F:M\times I\to M\times I$$F:M\times I\to M\times I$ such that

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

See [Skopenkov2006, Figure 1.1]. An ambient isotopy is the above homeomorphism $F$$F$, or, equivalently, a family of homeomorphisms $F_t:M\to M$$F_t:M\to M$ generated by the map $F$$F$ in the obvious manner. The latter family can be seen as a homotopy $M\times I\to M$$M\times I\to M$.

Evidently, ambient isotopy is an equivalence relation on the set of embeddings of $N$$N$ into $M$$M$. Classification of embeddings up to ambient isotopy is a classical problem in topology, see [Skopenkov2016c, $\S$$\S$1].

Ambient isotopy is a stronger equivalence relation than 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 or for homotopy in the class of embeddings.

Definition 2.2 (Non-ambient isotopy). For manifolds $M,N$$M,N$ two embeddings $f,g:N\to M$$f,g:N\to M$ are called non-ambient isotopic, if there exists an embedding $F:N\times I\to M\times I$$F:N\times I\to M\times I$ such that

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

This is equivalent to the existence of a homotopy in the class of embeddings. (Recall that we tacitly work with compact manifolds. For counterexample involving non-compact manifolds see [Geiges2018].)

In the smooth category, or for $m-n\ge3$$m-n\ge3$ in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975]. For $m-n\le2$$m-n\le2$ this is not so: e.g., any knot $S^1\to\Rr^3$$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 called diffeotopy.

3 Isoposition and concordance

Definition 3.1 (Isoposition). For manifolds $M,N$$M,N$ two embeddings $f,g:N\to M$$f,g:N\to M$ are called (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism $h:M\to M$$h:M\to M$ such that $h\circ f=g$$h\circ f=g$.

For embeddings into $\Rr^m$$\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.

Definition 3.2 (Concordance). For manifolds $M,N$$M,N$ two embeddings $f,g:N\to M$$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$$F:M\times I\to M\times I$ (which is called a concordance) such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for each $y\in M$$y\in M$ and
• $F(f(x),1)=(g(x),1)$$F(f(x),1)=(g(x),1)$ for each $x\in N$$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$$m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy [Lickorish1965], [Hudson1970], [Hudson&Lickorish1971]. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the Knotting Problem to the relativized Embedding Problem, see [Skopenkov2016c, $\S$$\S$1].

4 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
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.

]{Skopenkov2016c}. Ambient isotopy is a stronger equivalence relation than 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'' or for ''homotopy in the class of embeddings''. {{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}} This is equivalent to the existence of a ''homotopy in the class of embeddings''. (Recall that we tacitly work with compact manifolds. For counterexample involving non-compact manifolds see \cite{Geiges2018}.) In the smooth 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. In the smooth category, ''non-ambient isotopy'' is 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) 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[$\SM,N two embeddings $f,g:N\to M$$f,g:N\to M$ are called ambiently isotopic, if there exists a homeomorphism onto $F:M\times I\to M\times I$$F:M\times I\to M\times I$ such that

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