Isotopy

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Classification of embeddings up to isotopy is a classical problem in topology, see [Skopenkov2016c, $\S$${{Stub}} <wikitex>; [[High codimension embeddings: classification#Introduction|Classification of embeddings up to isotopy]] is a classical problem in topology, see \cite[$\S\S$1].

1 Definition of ambient isotopy

For manifolds $M,N$$M,N$ two embeddings $f,g:N\to M$$f,g:N\to M$ are called ambiently isotopic, if there exists a homeomorphism onto $F:\Rr^m\times I\to\Rr^m\times I$$F:\Rr^m\times I\to\Rr^m\times I$ such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for each $y\in\Rr^m,$$y\in\Rr^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(\Rr^m\times\{t\})=\Rr^m\times\{t\}$$F(\Rr^m\times\{t\})=\Rr^m\times\{t\}$ for each $t\in I.$$t\in I.$

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

Evidently, isotopy is an equivalence relation on the set of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$.

This notion of isotopy is also called ambient isotopy in contrast to the non-ambient isotopy defined just below.

2 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$$f,g:N\to\Rr^m$ are called non-ambient isotopic, if there exists an embedding $F:N\times I\to\Rr^m\times I$$F:N\times I\to\Rr^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$$x\in N$ and
• $F(N\times\{t\})\subset\Rr^m\times\{t\}$$F(N\times\{t\})\subset\Rr^m\times\{t\}$ for each $t\in I$$t\in I$.

In the DIFF 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.

Two embeddings $f,g:N\to\Rr^m$$f,g:N\to\Rr^m$ are said to be (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism $h:\Rr^m\to\Rr^m$$h:\Rr^m\to\Rr^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 ambient isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22].

Two embeddings $f,g:N\to\Rr^m$$f,g:N\to\Rr^m$ are said to be (ambiently) concordant if there is a homeomorphism (onto) $F:\Rr^m\times I\to\Rr^m\times I$$F:\Rr^m\times I\to\Rr^m\times I$ (which is called a concordance) such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for each $y\in\Rr^m$$y\in\Rr^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.

In the DIFF category or for $m-n\ge3$$m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy [Lickorish1965], [Hudson1970], [Hudson&Lickorish1971]. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the Knotting Problem to the relativized Embedding Problem, see [SkopenkovC, $\S$$\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
• [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.
• [SkopenkovC] A. Skopenkov, High codimension embeddings: classification, , submitted to Bull. MA

]{Skopenkov2016c}. == Definition of ambient isotopy == ; For manifolds $M,N$ two [[Embedding|embeddings]] $f,g:N\to M$ are called [[Wikipedia:Ambient_isotopy|ambiently 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 ''ambient isotopy'' is the above homeomorphism $F$, or, equivalently, a family of homeomorphisms $F_t:\Rr^m\to\Rr^m$ generated by the map $F$ in the obvious manner. The latter family can be seen as a homotopy $\Rr^m\times I\to\Rr^m$. 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[$\S\S1].

1 Definition of ambient isotopy

For manifolds $M,N$$M,N$ two embeddings $f,g:N\to M$$f,g:N\to M$ are called ambiently isotopic, if there exists a homeomorphism onto $F:\Rr^m\times I\to\Rr^m\times I$$F:\Rr^m\times I\to\Rr^m\times I$ such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for each $y\in\Rr^m,$$y\in\Rr^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(\Rr^m\times\{t\})=\Rr^m\times\{t\}$$F(\Rr^m\times\{t\})=\Rr^m\times\{t\}$ for each $t\in I.$$t\in I.$

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

Evidently, isotopy is an equivalence relation on the set of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$.

This notion of isotopy is also called ambient isotopy in contrast to the non-ambient isotopy defined just below.

2 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$$f,g:N\to\Rr^m$ are called non-ambient isotopic, if there exists an embedding $F:N\times I\to\Rr^m\times I$$F:N\times I\to\Rr^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$$x\in N$ and
• $F(N\times\{t\})\subset\Rr^m\times\{t\}$$F(N\times\{t\})\subset\Rr^m\times\{t\}$ for each $t\in I$$t\in I$.

In the DIFF 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.

Two embeddings $f,g:N\to\Rr^m$$f,g:N\to\Rr^m$ are said to be (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism $h:\Rr^m\to\Rr^m$$h:\Rr^m\to\Rr^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 ambient isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22].

Two embeddings $f,g:N\to\Rr^m$$f,g:N\to\Rr^m$ are said to be (ambiently) concordant if there is a homeomorphism (onto) $F:\Rr^m\times I\to\Rr^m\times I$$F:\Rr^m\times I\to\Rr^m\times I$ (which is called a concordance) such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for each $y\in\Rr^m$$y\in\Rr^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.

In the DIFF category or for $m-n\ge3$$m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy [Lickorish1965], [Hudson1970], [Hudson&Lickorish1971]. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the Knotting Problem to the relativized Embedding Problem, see [SkopenkovC, $\S$$\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
• [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.
• [SkopenkovC] A. Skopenkov, High codimension embeddings: classification, , submitted to Bull. MA

]{SkopenkovC}. == References == {{#RefList:}} [[Category:Definitions]] [[Category:Embeddings of manifolds]]\S1].

1 Definition of ambient isotopy

For manifolds $M,N$$M,N$ two embeddings $f,g:N\to M$$f,g:N\to M$ are called ambiently isotopic, if there exists a homeomorphism onto $F:\Rr^m\times I\to\Rr^m\times I$$F:\Rr^m\times I\to\Rr^m\times I$ such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for each $y\in\Rr^m,$$y\in\Rr^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(\Rr^m\times\{t\})=\Rr^m\times\{t\}$$F(\Rr^m\times\{t\})=\Rr^m\times\{t\}$ for each $t\in I.$$t\in I.$

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

Evidently, isotopy is an equivalence relation on the set of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$.

This notion of isotopy is also called ambient isotopy in contrast to the non-ambient isotopy defined just below.

2 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$$f,g:N\to\Rr^m$ are called non-ambient isotopic, if there exists an embedding $F:N\times I\to\Rr^m\times I$$F:N\times I\to\Rr^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$$x\in N$ and
• $F(N\times\{t\})\subset\Rr^m\times\{t\}$$F(N\times\{t\})\subset\Rr^m\times\{t\}$ for each $t\in I$$t\in I$.

In the DIFF 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.

Two embeddings $f,g:N\to\Rr^m$$f,g:N\to\Rr^m$ are said to be (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism $h:\Rr^m\to\Rr^m$$h:\Rr^m\to\Rr^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 ambient isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22].

Two embeddings $f,g:N\to\Rr^m$$f,g:N\to\Rr^m$ are said to be (ambiently) concordant if there is a homeomorphism (onto) $F:\Rr^m\times I\to\Rr^m\times I$$F:\Rr^m\times I\to\Rr^m\times I$ (which is called a concordance) such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for each $y\in\Rr^m$$y\in\Rr^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.

In the DIFF category or for $m-n\ge3$$m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy [Lickorish1965], [Hudson1970], [Hudson&Lickorish1971]. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the Knotting Problem to the relativized Embedding Problem, see [SkopenkovC, $\S$$\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
• [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.
• [SkopenkovC] A. Skopenkov, High codimension embeddings: classification, , submitted to Bull. MA