Isotopy
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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. | 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 also called ''diffeotopy''.--> | + | <!--In the smooth category, ''non-ambient isotopy'' is also called ''diffeotopy''.--> |
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Revision as of 12:57, 14 March 2019
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Contents |
1 Introduction
We work in the smooth or piecewise-linear (PL) or topological (TOP) categories. If a category is omitted, then the result holds (or a definition is given) in all the three categories. We write `CAT homeomorphism' to mean `diffeomorphism' for CAT=DIFF, `PL homemomorphism' for CAT=PL and `homeomorphism' for CAT=TOP. 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). All manifolds are 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 CAT homeomorphism such that
- for each
- for each and
- for each
See [Skopenkov2006, Figure 1.1].
This defines an equivalence relation on the set of embeddings of intoTex syntax error(in the smooth category this is not so trivial, see [Hirsch1976, 8, Theorem 1.9 and Excercise 1]).
This equivalence relation is called `ambient isotopy'.
For a simple example see [Skopenkov2016c, Remark 1.2.a]. The classification of embeddings up to ambient isotopy is a classical problem in topology, see [Skopenkov2016c] for introduction and fundamental theorems in which the concept plays a key role.
The above CAT homeomorphism , or the family (=the set) of CAT homeomorphisms defined by , are also called ambient isotopy.
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 sometimes stands for non-ambient isotopy.
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 .
For more delicate questions involving homotopy in the space of embeddings and non-compact manifolds see [Geiges2018].
Theorem 2.3. In the smooth category, or for in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976, 8.1], [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.
3 Isoposition and concordance
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].
This defines an equivalence relation on the set of embeddings of intoTex syntax error(in the smooth category this is not so trivial, see [Hirsch1976, 8, Theorem 1.9 and Excercise 1]).
This equivalence relation is called `ambient isotopy'.
For a simple example see [Skopenkov2016c, Remark 1.2.a]. The classification of embeddings up to ambient isotopy is a classical problem in topology, see [Skopenkov2016c] for introduction and fundamental theorems in which the concept plays a key role.
The above CAT homeomorphism , or the family (=the set) of CAT homeomorphisms defined by , are also called ambient isotopy.
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 sometimes stands for non-ambient isotopy.
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 .
For more delicate questions involving homotopy in the space of embeddings and non-compact manifolds see [Geiges2018].
Theorem 2.3. In the smooth category, or for in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976, 8.1], [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.
3 Isoposition and concordance
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.