Isotopy
(Created page with '<wikitex> Two embeddings $f,g:N\to\Rr^m$ are said to be $(ambient)$ isotopic (see \cite{Skopenkov2006}, Figure 1.1), if there exists a homeomorphism onto (an ambient isotopy) $F:…') |
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− | <wikitex> | + | == Definition == |
+ | <wikitex>; | ||
Two embeddings $f,g:N\to\Rr^m$ are said to be $(ambient)$ isotopic (see \cite{Skopenkov2006}, Figure 1.1), if there exists a homeomorphism onto (an ambient isotopy) $F:\Rr^m\times I\to\Rr^m\times I$ such that | Two embeddings $f,g:N\to\Rr^m$ are said to be $(ambient)$ isotopic (see \cite{Skopenkov2006}, Figure 1.1), if there exists a homeomorphism onto (an ambient 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 $y\in\Rr^m,$ | ||
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Evidently, isotopy is an equivalence relation on the set of embeddings of $N$ into $\Rr^m$. | Evidently, isotopy is an equivalence relation on the set of embeddings of $N$ into $\Rr^m$. | ||
+ | </wikitex> | ||
+ | |||
+ | == Other equivalence relations == | ||
+ | <wikitex>; | ||
+ | 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 isotopy implies ambient isotopy \cite{Hudson&Zeeman1964}, \cite{Hudson1966}, \cite{Akin1969}, \cite{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 isotopy (the Alexander-Guggenheim Theorem) \cite{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 concordance implies ambient concordance and isotopy \cite{Lickorish1965}, \cite{Hudson1970}, \cite{Hudson&Lickorish1971}. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the problem of isotopy to the relativized problem of embeddability. | ||
</wikitex> | </wikitex> |
Revision as of 16:44, 10 March 2010
1 Definition
Two embeddings are said to be isotopic (see [Skopenkov2006], Figure 1.1), if there exists a homeomorphism onto (an ambient isotopy) such that
- for each
- for each and
- for each
An ambient isotopy is also a homotopy or a family of homeomorphisms generated by the map in the obvious manner.
Evidently, isotopy is an equivalence relation on the set of embeddings of into .
2 Other equivalence relations
Ambient isotopy is a stronger equivalence relation than non-ambient isotopy, isoposition, concordance, bordism, etc.
Two embeddings are called isotopic, if there exists an embedding such that
- ,
- for each and
- for each .
In the DIFF category or for in the PL or TOP category isotopy implies ambient isotopy [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975], \S7. For this is not so: e.g., any knot is non-ambiently PL isotopic to the trivial one, but not necessarily ambiently PL isotopic to it.
Two embeddings are said to be (orientation preserving) , if there is an (orientation preserving) homeomorphism such that .
For embeddings into PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972], 3.22.
Two embeddings are said to be concordant if there is a homeomorphism (onto) (which is called a ) such that
- for each and
- for each .
The definition of - is analogously obtained from that of non-ambient isotopy by dropping the last condition of level-preservation.
In the DIFF category or for in the PL or TOP category concordance implies ambient concordance and 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 problem of isotopy to the relativized problem of embeddability.