Isotopy
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Contents |
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 two embeddings are called ambiently isotopic, if there exists a homeomorphism onto such that
- for each
- for each and
- for each
See [Skopenkov2006, Figure 1.1]. An ambient isotopy is the above homeomorphism , or, equivalently, a family of homeomorphisms generated by the map in the obvious manner. The latter family can be seen as a homotopy .
Evidently, ambient isotopy is an equivalence relation on the set of embeddings of intoTex syntax error.
Classification of embeddings up to ambient isotopy is a classical problem in topology, see [Skopenkov2016c, 1].
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 or for homotopy in the class of embeddings.
Definition 2.2 (Non-ambient isotopy). For manifolds two embeddings are called non-ambient isotopic, if there exists an embedding such that
- ,
- for each and
- for each .
This is equivalent to the existence of a homotopy in the space of embeddings. (Recall that we work with compact manifolds. For a counterexample involving non-compact manifolds see [Geiges2018].)
In the smooth category, or for in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975]. For this is not so: e.g., any knot 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.
3 Isoposition and concordance
We write `CAT homeomorphism' meaning diffeomorphism for CAT=DIFF, PL homemomrphism for CAT=DIFF and TOP homeomorphism (often called just `homeomorphism') for CAT=TOP. Here CAT coincides with the category (of manifolds and their maps) omitted elsewhere.
Definition 3.1 (Isoposition). For manifolds two embeddings are called (orientation preserving) isopositioned, if there is an (orientation preserving) CAT homeomorphism such that .
For embeddings into 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 two embeddings are called ambiently concordant, or just concordant, if there is a homeomorphism onto (which is called a concordance) such that
- for each and
- for each .
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 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, 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.
- for each
- for each and
- for each
See [Skopenkov2006, Figure 1.1]. An ambient isotopy is the above homeomorphism , or, equivalently, a family of homeomorphisms generated by the map in the obvious manner. The latter family can be seen as a homotopy .
Evidently, ambient isotopy is an equivalence relation on the set of embeddings of intoTex syntax error.
Classification of embeddings up to ambient isotopy is a classical problem in topology, see [Skopenkov2016c, 1].
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 or for homotopy in the class of embeddings.
Definition 2.2 (Non-ambient isotopy). For manifolds two embeddings are called non-ambient isotopic, if there exists an embedding such that
- ,
- for each and
- for each .
This is equivalent to the existence of a homotopy in the space of embeddings. (Recall that we work with compact manifolds. For a counterexample involving non-compact manifolds see [Geiges2018].)
In the smooth category, or for in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975]. For this is not so: e.g., any knot 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.
3 Isoposition and concordance
We write `CAT homeomorphism' meaning diffeomorphism for CAT=DIFF, PL homemomrphism for CAT=DIFF and TOP homeomorphism (often called just `homeomorphism') for CAT=TOP. Here CAT coincides with the category (of manifolds and their maps) omitted elsewhere.
Definition 3.1 (Isoposition). For manifolds two embeddings are called (orientation preserving) isopositioned, if there is an (orientation preserving) CAT homeomorphism such that .
For embeddings into 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 two embeddings are called ambiently concordant, or just concordant, if there is a homeomorphism onto (which is called a concordance) such that
- for each and
- for each .
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 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, 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.
- for each
- for each and
- for each
See [Skopenkov2006, Figure 1.1]. An ambient isotopy is the above homeomorphism , or, equivalently, a family of homeomorphisms generated by the map in the obvious manner. The latter family can be seen as a homotopy .
Evidently, ambient isotopy is an equivalence relation on the set of embeddings of intoTex syntax error.
Classification of embeddings up to ambient isotopy is a classical problem in topology, see [Skopenkov2016c, 1].
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 or for homotopy in the class of embeddings.
Definition 2.2 (Non-ambient isotopy). For manifolds two embeddings are called non-ambient isotopic, if there exists an embedding such that
- ,
- for each and
- for each .
This is equivalent to the existence of a homotopy in the space of embeddings. (Recall that we work with compact manifolds. For a counterexample involving non-compact manifolds see [Geiges2018].)
In the smooth category, or for in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975]. For this is not so: e.g., any knot 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.
3 Isoposition and concordance
We write `CAT homeomorphism' meaning diffeomorphism for CAT=DIFF, PL homemomrphism for CAT=DIFF and TOP homeomorphism (often called just `homeomorphism') for CAT=TOP. Here CAT coincides with the category (of manifolds and their maps) omitted elsewhere.
Definition 3.1 (Isoposition). For manifolds two embeddings are called (orientation preserving) isopositioned, if there is an (orientation preserving) CAT homeomorphism such that .
For embeddings into 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 two embeddings are called ambiently concordant, or just concordant, if there is a homeomorphism onto (which is called a concordance) such that
- for each and
- for each .
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 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, 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.