Isotopy

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We work in the smooth or piecewise-linear (PL) or topological (TOP) category If a category is omitted, then the result holds (or a definition is given) in all the three categories.

All manifolds are tacitly assumed to be compact. For examples involving non-compact manifolds see e.g. [Geiges2018].

1 Ambient and non-ambient isotopy

Definition 1.1 (Ambient isotopy). For manifolds $M,N$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M,N$ two embeddings $f,g:N\to M$$f,g:N\to M$ are called ambiently isotopic, if there exists a homeomorphism onto $F:M\times I\to M\times I$$F:M\times I\to M\times I$ such that

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

See [Skopenkov2006, Figure 1.1]. An ambient isotopy is the above homeomorphism $F$$F$, or, equivalently, a family of homeomorphisms $F_t:M\to M$$F_t:M\to M$ generated by the map $F$$F$ in the obvious manner. The latter family can be seen as a homotopy $M\times I\to M$$M\times I\to M$.

Evidently, ambient isotopy is an equivalence relation on the set of embeddings of $N$$N$ into $M$$M$. Classification of embeddings up to ambient isotopy is a classical problem in topology, see [Skopenkov2016c, $\S$$\S$1].

Ambient isotopy is a stronger equivalence relation than 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 often stands for non-ambient isotopy or for homotopy in the class of embeddings.

Definition 1.2 (Non-ambient isotopy). For manifolds $M,N$$M,N$ two embeddings $f,g:N\to M$$f,g:N\to M$ are called non-ambient 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 each $x\in N$$x\in N$ and
• $F(N\times\{t\})\subset M\times\{t\}$$F(N\times\{t\})\subset M\times\{t\}$ for each $t\in I$$t\in I$.

This is equivalent to the existence of a homotopy in the class of embeddings.

In the smooth category, or for $m-n\ge3$$m-n\ge3$ in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975]. For $m-n\le2$$m-n\le2$ this is not so: e.g., any knot $S^1\to\Rr^3$$S^1\to\Rr^3$ is non-ambiently PL isotopic to the trivial one, but not necessarily ambiently PL isotopic to it.

In the smooth category, non-ambient isotopy is called diffeotopy.

2 Isoposition and concordance

Definition 2.1 (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) 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]. Itw ould be interesting to know if the smooth analogue of this result holds.

Definition 2.2 (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].

3 References

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• [Edwards1975] R. D. Edwards, The equivalence of close piecewise linear embeddings, General Topol. Appl. 5 (1975), 147–180. MR0370603 (51 #6830) Zbl 0314.57009
• [Geiges2018] H. Geiges, Isotopies vis-`a-vis level-preserving embeddings, Arch. Math. 110 (2018), 197–200. Available at the arXiv:1708.09703.
• [Hirsch1976] M. W. Hirsch, Differential topology., Graduate Texts in Mathematics, No. 33. Springer-Verlag., New York-Heidelberg, 1976. MR0448362 (56 #6669) Zbl 0356.57001
• [Hudson&Lickorish1971] J. F. P. Hudson and W. B. R. Lickorish, Extending piecewise linear concordances, Quart. J. Math. Oxford Ser. (2) 22 (1971), 1–12. MR0290373 (44 #7557) Zbl 0219.57011
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• [Hudson1970] J. F. P. Hudson, Concordance, isotopy, and diffeotopy, Ann. of Math. (2) 91 (1970), 425–448. MR0259920 (41 #4549) Zbl 0202.54602
• [Lickorish1965] W. B. R. Lickorish, The piecewise linear unknotting of cones, Topology 4 (1965), 67–91. MR0203736 (34 #3585) Zbl 0138.19003
• [Rourke&Sanderson1972] C. Rourke and B. Sanderson, Introduction to piecewise-linear topology., Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 69. Berlin-Heidelberg-New York: Springer-Verlag. VIII, 1972. MR0665919 (83g:57009) Zbl 0254.57010
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.