Isotopy
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In the DIFF category or for $m-n\ge3$ in the PL or TOP category concordance implies ambient concordance and 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 problem of isotopy to the relativized problem of embeddability. | In the DIFF category or for $m-n\ge3$ in the PL or TOP category concordance implies ambient concordance and 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 problem of isotopy to the relativized problem of embeddability. | ||
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Revision as of 16:48, 10 March 2010
1 Definition
Two embeddings are said to be () isotopic (see [Skopenkov2006], Figure 1.1), if there exists a homeomorphism onto (an ambient isotopy) such that
- for each
- for each and
- for each
An ambient isotopy is also 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 .
2 Other equivalence relations
Ambient isotopy is a stronger equivalence relation than non-ambient isotopy, isoposition, concordance, bordism, etc.
Two embeddings are called (-) 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 isotopy implies ambient isotopy [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975], \S7. 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) , 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.
Two embeddings are said to be concordant if there is a homeomorphism (onto) (which is called a ) such that
- for each and
- for each .
The definition of - 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 concordance implies ambient concordance and 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 problem of isotopy to the relativized problem of embeddability.
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
- [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.
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