Isotopy

Isotopy
1 Definition
For a space N two embeddings f,g:N→R m are said to be isotopic, if there exists a homeomorphism onto F:R m×I→R m×I such that
∙F(y,0)=(y,0) for each y∈R m,
∙F(f(x),1)=(g(x),1) for each x∈N, and
∙F(R m×{t})=R m×{t} for each t∈I.
See e.g. [Skopenkov2006], Figure 1.1. An isotopy is the above homeomorphism F, or, equivalently, a homotopy R m×I→R m or a family of homeomorphisms F t:R m→R m generated by the map F in the obvious manner.
Evidently, isotopy is an equivalence relation on the set of embeddings of N into R m.
This notion of isotopy is also called ambient isotopy in contrast to the non-ambient isotopy defined just below.
[edit] 2 Other equivalence relations
Ambient isotopy is a stronger equivalence relation than non-ambient isotopy, isoposition, concordance, bordism, etc.
Two embeddings f,g:N→R m are called non-ambient isotopic, if there exists an embedding F:N×I→R m×I such that
∙F(x,0)=(f(x),0),
∙F(x,1)=(g(x),1) for each x∈N and
∙F(N×{t})⊂R m×{t} for each t∈I.
In the DIFF category or for m−n≥3in the PL or TOP category non-ambient isotopy implies ambient isotopy [Hirsch1976], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975], \S7. For m−n≤2this is not so: e.g., any knot S1→R3 is non-ambiently PL isotopic to the trivial one, but not necessarily ambiently PL isotopic to it.
Two embeddings f,g:N→R m are said to be (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism h:R m→R m such that h∘f=g.
For embeddings into R m PL orientation preserving isoposition is equivalent to PL ambient isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972], 3.22.
Two embeddings f,g:N→R m are said to be (ambiently) concordant if there is a homeomorphism (onto) F:R m×I→R m×I(which is called a concordance) such that
∙F(y,0)=(y,0) for each y∈R m and
∙F(f(x),1)=(g(x),1) for each x∈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≥3in 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.。