Isotopy

Definition 1.1 (Ambient isotopy). For manifolds ${{Authors|Askopenkov}} ==Definition== <wikitex>; We work in the [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] or [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) or [[Wikipedia:Topological_manifold|topological]] (TOP) categories. If a category is omitted, then the result holds (or a definition is given) in all the three categories. We write `CAT homeomorphism' to mean `diffeomorphism' for CAT=DIFF, `PL homemomorphism' for CAT=PL and `homeomorphism' for CAT=TOP. Here CAT coincides with the category (of manifolds and their maps), which is omitted elsewhere in the sentence involving `CAT homeomorphism'. By a homeomorphism we mean a homeomorphism onto (as opposed to an embedding). All manifolds are assumed to be compact. {{beginthm|Definition|(Ambient isotopy)}}\label{dai} For manifolds $M,N$ two [[Embedding_(simple_definition)|embeddings]] $f,g:N\to M$ are called [[Wikipedia:Ambient_isotopy|ambiently isotopic]], if there exists a CAT homeomorphism $F:M\times I\to M\times I$ 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 [[Media:1-1.pdf|the figure]] for $M=\Rr^m$.<!--\cite[Figure 1.1]{Skopenkov2006}--> This defines an equivalence relation on the set of embeddings of $N$ into $M$ (in the smooth category this is not so trivial, see \cite[$\S, Theorem 1.9]{Hirsch1976}). This equivalence relation is called `ambient isotopy'. {{endthm}} For a simple example see \cite[Remark 1.3.a]{Skopenkov2016c}. The [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|classification of embeddings up to ambient isotopy]] is a classical problem in topology, see \cite{Skopenkov2016c} for introduction and fundamental theorems in which the concept plays a key role. The above CAT homeomorphism $F$, or the family (=the set) of CAT homeomorphisms $F_t:M\to M$ defined by $F_t(x):=F(t,x)$, are also called an ''ambient isotopy''. <!--The latter family can be seen as a homotopy $M\times I\to M$.--> Ambient isotopy is a stronger equivalence relation than any of the relations 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'' sometimes stands for ''non-ambient isotopy''. <!-- or for ''homotopy in the class of embeddings''.--> {{beginthm|Definition|(Non-ambient isotopy)}}\label{di} 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 defines an equivalence relation on the set of embeddings of $N$ into $M$ (in the smooth category this is not so trivial, see \cite[$\S, Theorem 1.9 and Excercise 1]{Hirsch1976}). This equivalence relation is called `ambient isotopy'. {{endthm}} For more delicate questions involving ''homotopy in the space of embeddings'' and non-compact manifolds see \cite{Geiges2018}. {{beginthm|Theorem}}\label{t:aivsi} In the smooth category, or for $m-n\ge3$ in the PL or TOP category, non-ambient isotopy implies ambient isotopy \cite[$\S.1]{Hirsch1976}, \cite{Hudson&Zeeman1964}, \cite{Hudson1966}, \cite{Akin1969}, \cite{Edwards1975}. {{endthm}} 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 also called ''diffeotopy''.--> </wikitex> == Isoposition and concordance == <wikitex>; {{beginthm|Definition|(Isoposition)}}\label{dipn} For manifolds $M,N$ two embeddings $f,g:N\to M$ are called (orientation preserving) ''isopositioned'', if there is an (orientation preserving) CAT homeomorphism $h:M\to M$ such that $h\circ f=g$. {{endthm}} For embeddings into $\Rr^m$ PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) \cite[3.22]{Rourke&Sanderson1972}. It would be interesting to know if the smooth analogue of this result holds. {{beginthm|Definition|(Concordance)}}\label{dac} 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 $x\in N$. {{endthm}} 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 \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 [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Knotting Problem]] to the relativized [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Embedding Problem]], see \cite[$\SM,N$ an ambient isotopy between two embeddings $f,g:N\to M$ is a CAT homeomorphism $F:M\times I\to M\times I$ such that

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

See the figure for $M=\Rr^m$. This defines an equivalence relation on the set of embeddings of $N$ into $M$ (in the smooth category this is non-trivial, see [Hirsch1976, $\S$8, Theorem 1.9]). This equivalence relation is called `ambient isotopy'.

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 $t\in I$.

This defines an equivalence relation on the set of embeddings of $N$ into $M$ (in the smooth category this is not so trivial, see [Hirsch1976, $\S$8, Theorem 1.9 and Excercise 1]). This equivalence relation is called `ambient isotopy'.

Theorem 1.3. In the smooth category, or for $m-n\ge3$ in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976, $\S$8.1], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975].

Definition 2.1 (Isoposition). For manifolds $M,N$ two embeddings $f,g:N\to M$ are called (orientation preserving) isopositioned, if there is an (orientation preserving) CAT homeomorphism $h:M\to M$ such that $h\circ f=g$.

Definition 2.2 (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 $x\in N$.