Isotopy

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== Definition ==
== Definition ==
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Two embeddings $f,g:N\to\Rr^m$ are said to be isotopic (see \cite{Skopenkov2006}, Figure 1.1), if there exists a homeomorphism onto (an isotopy) $F:\Rr^m\times I\to\Rr^m\times I$ such that
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Two embeddings $f,g:N\to\Rr^m$ are said to be [[Wikipedia:Ambient_isotopy|isotopic]] (see \cite{Skopenkov2006}, Figure 1.1), if there exists a homeomorphism onto (an isotopy) $F:\Rr^m\times I\to\Rr^m\times I$ such that
* $F(y,0)=(y,0)$ for each $y\in\Rr^m,$
* $F(y,0)=(y,0)$ for each $y\in\Rr^m,$
* $F(f(x),1)=(g(x),1)$ for each $x\in N,$ and
* $F(f(x),1)=(g(x),1)$ for each $x\in N,$ and

Revision as of 12:26, 20 March 2010

Classification of embeddings up to isotopy is a classical problem in topology.

1 Definition

Two embeddings f,g:N\to\Rr^m are said to be isotopic (see [Skopenkov2006], Figure 1.1), if there exists a homeomorphism onto (an isotopy) F:\Rr^m\times I\to\Rr^m\times I such that

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

An isotopy is also a homotopy \Rr^m\times I\to\Rr^m or a family of homeomorphisms F_t:\Rr^m\to\Rr^m generated by the map F in the obvious manner.

Evidently, isotopy is an equivalence relation on the set of embeddings of N into \Rr^m.

This notion of isotopy is also called ambient isotopy in contrast to the non-ambient isotopy defined just below.

2 Other equivalence relations

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

Two embeddings f,g:N\to\Rr^m are called non-ambient isotopic, if there exists an embedding F:N\times I\to\Rr^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\Rr^m\times\{t\} for each t\in I.

In the DIFF category or for m-n\ge3 in the PL or TOP category non-ambient isotopy implies ambient isotopy [Hirsch1976], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975], \S7. 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.

Two embeddings f,g:N\to\Rr^m are said to be (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism h:\Rr^m\to\Rr^m such that h\circ f=g.

For embeddings into \Rr^m PL orientation preserving isoposition is equivalent to PL ambient isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972], 3.22.

Two embeddings f,g:N\to\Rr^m are said to be (ambiently) concordant if there is a homeomorphism (onto) F:\Rr^m\times I\to\Rr^m\times I (which is called a concordance) such that

  • F(y,0)=(y,0) for each y\in\Rr^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.

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.

3 References

This page has not been refereed. The information given here might be incomplete or provisional.

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