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 I = [0,1] denotes the unit interval.

Definition 1.1 (Ambient isotopy). For manifolds M,N an ambient isotopy between two CAT embeddings f,g:N\to M is a CAT homeomorphism F:M\times I\to M\times I such that

  • F(y,0)=(y,0) for all y\in M,
  • F(f(x),1)=(g(x),1) for all x\in N, and
  • F(M\times\{t\})=M\times\{t\} for all t \in I.
An ambient isotopy for M=\Rr^m: the picture is realistic for N = S^1 and M = \R^2

Two embeddings f and g 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 N into M (in the smooth category this is non-trivial and proven in [Hirsch1976, \S8, 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 N = \R^m and also a summary of theorems stating when all embeddings M \to \R^m 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 M,N two CAT embeddings f,g:N\to M are called CAT isotopic, if there exists an embedding F:N\times I\to M\times I such that

  • F(x,0)=(f(x),0),
  • F(x,1)=(g(x),1) for all x\in N and
  • F(N\times\{t\})\subset M\times\{t\} for all t\in I.

Two embeddings f and g are called isotopic if there is an isotopy between them. Isotopy defines an equivalence relation on the set of embeddings of N into M (in the smooth category this is non-trivial, see [Hirsch1976, \S8, 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 M into N 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 M into N coincides with the path components of the space of embeddings of M into N. 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 m-n\ge3 in the PL or TOP category, isotopy implies ambient isotopy [Hirsch1976, \S8.1], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975].

For m-n\le2 this is not so: e.g., any knot S^1\to\Rr^3 is PL isotopic to the unknot, but is not necessarily PL ambient isotopic to the unkot.

Definition 1.6 (Isoposition). For manifolds M,N two embeddings f,g:N\to M are called (orientation preserving) isopositioned, if there is an (orientation preserving) CAT homeomorphism h:M\to M such that h\circ f=g.

For embeddings into \Rr^m 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 M,N two embeddings f,g:N\to M are called ambiently concordant, or just concordant, if there is a homeomorphism onto F:M\times I\to M\times I (which is called a concordance) such that

  • F(y,0)=(y,0) for each y\in M and
  • F(f(x),1)=(g(x),1) for each x\in N.

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 m-n\ge3 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, \S1].

3 References

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