Isotopy

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Contents

[edit] 1 Introduction

We work in the smooth or piecewise-linear (PL) or topological (TOP) category. If a category is omitted, then the result holds (or a definition is given) in all the three categories.

All manifolds are tacitly assumed to be compact.

[edit] 2 Ambient and non-ambient isotopy

Definition 2.1 (Ambient isotopy). For manifolds M,N two embeddings f,g:N\to M are called ambiently isotopic, if there exists a homeomorphism onto F:M\times I\to M\times I such that

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

See [Skopenkov2006, Figure 1.1]. An ambient isotopy is the above homeomorphism F, or, equivalently, a family of homeomorphisms F_t:M\to M generated by the map F in the obvious manner. The latter family can be seen as a homotopy M\times I\to M.

Evidently, ambient isotopy is an equivalence relation on the set of embeddings of N into M. Classification of embeddings up to ambient isotopy is a classical problem in topology, see [Skopenkov2016c, \S1].

Ambient isotopy is a stronger equivalence relation than non-ambient isotopy, isoposition, concordance, bordism, etc., see below

The words ambient isotopy are often abbreviated to just isotopy. One should be careful because isotopy often stands for non-ambient isotopy or for homotopy in the class of embeddings.

Definition 2.2 (Non-ambient isotopy). For manifolds M,N two embeddings f,g:N\to M are called non-ambient 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 each x\in N and
  • F(N\times\{t\})\subset M\times\{t\} for each t\in I.

This is equivalent to the existence of a homotopy in the class of embeddings. (Recall that we tacitly work with compact manifolds. For counterexample involving non-compact manifolds see [Geiges2018].)

In the smooth category, or for m-n\ge3 in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975]. For m-n\le2 this is not so: e.g., any knot S^1\to\Rr^3 is non-ambiently PL isotopic to the trivial one, but not necessarily ambiently PL isotopic to it.

In the smooth category, non-ambient isotopy is called diffeotopy.

[edit] 3 Isoposition and concordance

Definition 3.1 (Isoposition). For manifolds M,N two embeddings f,g:N\to M are called (orientation preserving) isopositioned, if there is an (orientation preserving) 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.

Definition 3.2 (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].

[edit] 4 References

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