# Isotopy

## 1 Definition

This page defines isotopies and ambient isotopies between embeddings in either the smooth (DIFF), piecewise-linear (PL) or topological (TOP) categories. These notions usually appear in discussions of details, so a reader is more likely to see in the literature (including Manifold Atlas) isotopy and ambient isotopy as equivalence relations, which are also defined here. By a CAT embedding' we mean either a smooth embedding', a piecewise linear' embedding or a topological embedding', depending upon the category. By a CAT homeomorphism' we mean a diffeomorphism' if CAT=DIFF, a PL homemomorphism' if CAT=PL or ahomeomorphism' if CAT=TOP. All manifolds are assumed to be compact and $I = [0,1]$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}I = [0,1]$ denotes the unit interval.

Definition 1.1 (Ambient isotopy). For manifolds $M,N$$M,N$ an ambient isotopy between two CAT embeddings $f,g:N\to M$$f,g:N\to M$ is a CAT homeomorphism $F:M\times I\to M\times I$$F:M\times I\to M\times I$ such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for all $y\in M,$$y\in M,$
• $F(f(x),1)=(g(x),1)$$F(f(x),1)=(g(x),1)$ for all $x\in N,$$x\in N,$ and
• $F(M\times\{t\})=M\times\{t\}$$F(M\times\{t\})=M\times\{t\}$ for all $t \in I.$$t \in I.$
An ambient isotopy for $M=\Rr^m$$M=\Rr^m$: the picture is realistic for $N = S^1$$N = S^1$ and $M = \R^2$$M = \R^2$

Two embeddings $f$$f$ and $g$$g$ are called ambient isotopic if there is an ambient isotopy between them. Ambient isotopy defines an equivalence relation on the set of CAT embeddings of $N$$N$ into $M$$M$ (in the smooth category this is non-trivial and proven in [Hirsch1976, $\S$$\S$8, Theorem 1.9]).

For simple examples of ambient isotopic embeddings and also embeddings which are not ambient isotopic, see [Skopenkov2016c, Remark 1.3.a]. The classification of embeddings up to ambient isotopy is a classical problem in topology. For an introduction to the case when $N = \R^m$$N = \R^m$ and also a summary of theorems stating when all embeddings $M \to \R^m$$M \to \R^m$ are isotopic, see [Skopenkov2016c].

Remark 1.2. Some authors abbreviate ambient isotopy to just isotopy. Readers should be careful to clarify the meaning of isotopy in a particular text.

Definition 1.3 (Isotopy). For manifolds $M,N$$M,N$ two CAT embeddings $f,g:N\to M$$f,g:N\to M$ are called CAT isotopic, if there exists an embedding $F:N\times I\to M\times I$$F:N\times I\to M\times I$ such that

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

Two embeddings $f$$f$ and $g$$g$ are called isotopic if there is an isotopy between them. Isotopy defines an equivalence relation on the set of embeddings of $N$$N$ into $M$$M$ (in the smooth category this is non-trivial, see [Hirsch1976, $\S$$\S$8, Theorem 1.9 and Excercise 1]).

Remark 1.4. In the smooth category isotopy is also called diffeotopy by some authors.

The set of embeddings of $M$$M$ into $N$$N$ can be topologised in such a way that an isotopy is equivalent to a continuous path of embeddings. In this case the set of isotopy classes of embeddings of $M$$M$ into $N$$N$ coincides with the path components of the space of embeddings of $M$$M$ into $N$$N$. For details on the space of embeddings and for information in the case of non-compact manifolds see [Geiges2018].

Theorem 1.5. In the smooth category, or for $m-n\ge3$$m-n\ge3$ in the PL or TOP category, isotopy implies ambient isotopy [Hirsch1976, $\S$$\S$8.1], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975].

For $m-n\le2$$m-n\le2$ this is not so: e.g., any knot $S^1\to\Rr^3$$S^1\to\Rr^3$ is PL isotopic to the unknot, but is not necessarily PL ambient isotopic to the unkot.

Definition 1.6 (Isoposition). For manifolds $M,N$$M,N$ two embeddings $f,g:N\to M$$f,g:N\to M$ are called (orientation preserving) isopositioned, if there is an (orientation preserving) CAT homeomorphism $h:M\to M$$h:M\to M$ such that $h\circ f=g$$h\circ f=g$.

For embeddings into $\Rr^m$$\Rr^m$ PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22]. It would be interesting to know if the smooth analogue of this result holds.

## 2 Concordance

Definition 2.1 (Concordance). For manifolds $M,N$$M,N$ two embeddings $f,g:N\to M$$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$$F:M\times I\to M\times I$ (which is called a concordance) such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for each $y\in M$$y\in M$ and
• $F(f(x),1)=(g(x),1)$$F(f(x),1)=(g(x),1)$ for each $x\in N$$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$$m-n\ge3$ in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy [Lickorish1965], [Hudson1970], [Hudson&Lickorish1971]. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the Knotting Problem to the relativized Embedding Problem, see [Skopenkov2016c, $\S$$\S$1].