Isotopy
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Classification of embeddings up to isotopy is a classical problem in topology, see [Skopenkov2016c, 1].
1 Definition
For a space two embeddings are said to be isotopic, if there exists a homeomorphism onto such that
- for each
- for each and
- for each
See [Skopenkov2006, Figure 1.1]. An isotopy is the above homeomorphism , or, equivalently, a homotopy or a family of homeomorphisms generated by the map in the obvious manner.
Evidently, isotopy is an equivalence relation on the set of embeddings of into .
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 are called non-ambient isotopic, if there exists an embedding such that
- ,
- for each and
- for each .
In the DIFF 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.
Two embeddings are said to be (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism such that .
For embeddings into PL orientation preserving isoposition is equivalent to PL ambient isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22].
Two embeddings are said to be (ambiently) 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.
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 [SkopenkovC, 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
1 Definition
For a space two embeddings are said to be isotopic, if there exists a homeomorphism onto such that
- for each
- for each and
- for each
See [Skopenkov2006, Figure 1.1]. An isotopy is the above homeomorphism , or, equivalently, a homotopy or a family of homeomorphisms generated by the map in the obvious manner.
Evidently, isotopy is an equivalence relation on the set of embeddings of into .
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 are called non-ambient isotopic, if there exists an embedding such that
- ,
- for each and
- for each .
In the DIFF 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.
Two embeddings are said to be (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism such that .
For embeddings into PL orientation preserving isoposition is equivalent to PL ambient isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22].
Two embeddings are said to be (ambiently) 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.
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 [SkopenkovC, 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
1 Definition
For a space two embeddings are said to be isotopic, if there exists a homeomorphism onto such that
- for each
- for each and
- for each
See [Skopenkov2006, Figure 1.1]. An isotopy is the above homeomorphism , or, equivalently, a homotopy or a family of homeomorphisms generated by the map in the obvious manner.
Evidently, isotopy is an equivalence relation on the set of embeddings of into .
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 are called non-ambient isotopic, if there exists an embedding such that
- ,
- for each and
- for each .
In the DIFF 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.
Two embeddings are said to be (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism such that .
For embeddings into PL orientation preserving isoposition is equivalent to PL ambient isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22].
Two embeddings are said to be (ambiently) 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.
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 [SkopenkovC, 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