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This page defines isotopies and ambient isotopies between embeddings in either the smooth (DIFF), piecewise-linear (PL) or topological (TOP) categories. These notions usually appear in discussions of details, so a reader is more likely to see in the literature (including Manifold Atlas) isotopy and ambient isotopy as equivalence relations, which are also defined here. By a `CAT embedding' we mean either a `smooth embedding', a `piecewise linear' embedding or a `topological embedding', depending upon the category. By a `CAT homeomorphism' we mean a `diffeomorphism' if CAT=DIFF, a `PL homemomorphism' if CAT=PL or a`homeomorphism' if CAT=TOP. All manifolds are assumed to be compact and denotes the unit interval.
- for all
- for all and
- for all
Two embeddings and are called ambient isotopic if there is an ambient isotopy between them. Ambient isotopy defines an equivalence relation on the set of CAT embeddings of into (in the smooth category this is non-trivial and proven in [Hirsch1976, 8, Theorem 1.9]).
For simple examples of ambient isotopic embeddings and also embeddings which are not ambient isotopic, see [Skopenkov2016c, Remark 1.3.a]. The classification of embeddings up to ambient isotopy is a classical problem in topology. For an introduction to the case when and also a summary of theorems stating when all embeddings are isotopic, see [Skopenkov2016c].
Remark 1.2. Some authors abbreviate ambient isotopy to just isotopy. Readers should be careful to clarify the meaning of isotopy in a particular text.
Definition 1.3 (Isotopy). For manifolds two CAT embeddings are called CAT isotopic, if there exists an embedding such that
- for all and
- for all .
Two embeddings and are called isotopic if there is an isotopy between them. Isotopy defines an equivalence relation on the set of embeddings of into (in the smooth category this is non-trivial, see [Hirsch1976, 8, Theorem 1.9 and Excercise 1]).
Remark 1.4. In the smooth category isotopy is also called diffeotopy by some authors.
The set of embeddings of into can be topologised in such a way that an isotopy is equivalent to a continuous path of embeddings. In this case the set of isotopy classes of embeddings of into coincides with the path components of the space of embeddings of into . For details on the space of embeddings and for information in the case of non-compact manifolds see [Geiges2018].
For this is not so: e.g., any knot is PL isotopic to the unknot, but is not necessarily PL ambient isotopic to the unkot.
Definition 1.6 (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 2.1 (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].
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