Isotopy

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[edit] 1 Definitions

Definition 1.1 (Ambient isotopy, or just isotopy). For manifolds M,N two embeddings f,g:N\to M are called ambiently isotopic, or just 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, or just 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, isotopy is an equivalence relation on the set of embeddings of N into M. Classification of embeddings up to isotopy is a classical problem in topology, see [Skopenkov2016c, \S1].

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

Definition 1.2 (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.

In the DIFF category, or for m-n\ge3 in the PL or TOP category, non-ambient isotopy implies 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 PL isotopic to it.

Definition 1.3 (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]. Itw ould be interesting to know if the smooth analogue of this result holds.

Definition 1.4 (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] 2 References

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