Isotopy
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[edit] 1 Definitions
Definition 1.1 (Ambient isotopy, or just isotopy). For manifolds two embeddings are called ambiently isotopic, or just isotopic, if there exists a homeomorphism onto such that
- for each
- for each and
- for each
See [Skopenkov2006, Figure 1.1]. An ambient isotopy, or just 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, isotopy is an equivalence relation on the set of embeddings of into . Classification of embeddings up to isotopy is a classical problem in topology, see [Skopenkov2016c, 1].
Isotopy is a stronger equivalence relation than non-ambient isotopy, isoposition, concordance, bordism, etc.
Definition 1.2 (Non-ambient isotopy). For manifolds two embeddings are called non-ambient isotopic, if there exists an embedding such that
- ,
- for each and
- for each .
In the DIFF category, or for in the PL or TOP category, non-ambient isotopy implies 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 PL isotopic to it.
Definition 1.3 (Isoposition). For manifolds two embeddings are called (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism such that .
For embeddings into PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22]. Itw ould be interesting to know if the smooth analogue of this result holds.
Definition 1.4 (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].
[edit] 2 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
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- [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, High codimension embeddings: classification, submitted to Bull. Man. Atl..