# Isotopy

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## 1 Definition

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.

**Definition 1.1** (Ambient isotopy)**.**
For manifolds an ambient isotopy between two CAT embeddings is a CAT homeomorphism such that

- 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].

**Theorem 1.5.** In the smooth category, or for in the PL or TOP category, isotopy implies ambient isotopy [Hirsch1976, 8.1], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975].

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.

## 2 Concordance

**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].

## 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 - [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 - [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.