View source for Isotopy (in topology)

''This term may also have other meanings, see [[Isotopy]]''.

A [[Homotopy|homotopy]] of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i0529401.png" /> with respect to a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i0529402.png" /> is a continuous family 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i0529403.png" />   of [[Homeomorphism|homeomorphism]]  ( where, here and throughout, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i0529404.png" />)  from  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i0529407.png" /> onto a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i0529408.png" />. Analogously, an isotopy is a fibrewise-continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i0529409.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294010.png" /> takes the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294011.png" /> homeomorphically onto a subset of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294012.png" />. An isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294013.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294014.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294015.png" /> is called an isotopy of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294016.png" />.

By a covering (or enveloping) isotopy for an isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294017.png" /> one understands an isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294019.png" />. Two imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294021.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294022.png" /> are said to be isotopic if there exist a covering isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294023.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294025.png" />. Two spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294027.png" /> are said to be isotopy equivalent if there exist imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294029.png" /> such that the composites <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294031.png" /> are isotopic to the identity mapping. Isotopy-equivalent spaces are also called spaces of the same isotopy type (by analogy with the homotopy type). If two spaces are homeomorphic, then they are isotopy equivalent; however, there are non-homeomorphic spaces of the same isotopy type, for example, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294032.png" />-dimensional ball and the same ball with a line segment glued to it (at its ends). Any homotopy invariant is an isotopy invariant, but there exist isotopy invariants, for example dimension, that are not homotopy invariants.

The fundamental problem in isotopy theory is the isotopy extension problem, that is, the problem of the existence of an isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294033.png" /> covering a given isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294034.png" />. This problem, like the general problem of finding a complete system of isotopy invariants for imbeddings, is most often considered in the category of topological manifolds and its subcategories of piecewise-linear or differentiable manifolds.

A topological isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294035.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294036.png" /> with respect to a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294038.png" />, can be extended to a covering isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294039.png" /> if and only if the corresponding fibrewise imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294040.png" /> is locally flat (cf. [[Locally flat imbedding|Locally flat imbedding]]); here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294041.png" /> is a subinterval of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294042.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294044.png" />, then the isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294045.png" /> is covered by an isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294046.png" /> provided that the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294047.png" /> is locally flat for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294048.png" />. This is not true in codimension 2 (for example,  "tying a knot"  on a circle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294049.png" />), and therefore additional homotopy hypotheses are required for the existence of a covering isotopy. Sufficiently locally close flat imbeddings are isotopic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294050.png" />.

The existence theorem for a piecewise-linear isotopy is formulated in similar fashion in the general case (under the natural condition that the corresponding fibrewise imbeddings be locally flat in the piecewise-linear sense). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294051.png" />, then a piecewise-linear isotopy can always be extended, since in these codimensions a piecewise-linear imbedding is locally flat in the piecewise-linear sense. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294052.png" /> or 1 it is necessary to additionally suppose that the imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294053.png" /> be locally flat in the piecewise-linear sense, since in these codimensions, a piecewise-linear imbedding need not to be locally flat even in the topological sense, for example, a cone over a knot.

A differentiable isotopy can always be extended to a differentiable covering isotopy.

The problem of finding a complete system of isotopy invariants for imbeddings of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294054.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294055.png" /> has only been solved in a few particular cases. Thus, every locally flat (in the topological sense) imbedding of spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294056.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294057.png" /> or for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294059.png" />, is isotopic to the standard sphere, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294061.png" />, then a locally flat sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294063.png" /> is isotopic to the standard sphere if and only if the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294064.png" /> has the homotopy type of the circle (the Stallings–Brown theorem). In codimension 2 there can exist non-isotopic knots. In precisely the same way the Zeeman–Stallings theorem on the piecewise-linear isotopy (unknottedness) of piecewise-linear spheres can be stated.

An invariant of piecewise-linear isotopy is finer than an invariant of topological isotopy. Thus, the problem of topological isotopy of an arbitrary homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294065.png" /> to the identity has been solved in the positive sense for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294066.png" />, and under the same restriction the isotopy has been proved of any two orientation-preserving homeomorphism of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294067.png" /> onto itself. In the piecewise-linear case, these statements can be proved by elementary methods without any restrictions. Sufficiently close homeomorphisms of a topological manifold onto itself are isotopic; on the other hand, there exist arbitrarily close, piecewise-linear non-isotopic, piecewise-linear homeomorphisms of piecewise-linear manifolds, for example, of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294068.png" />-dimensional tori for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294069.png" />.

In contrast with the topological and piecewise-linear cases, it is by no means true that any two diffeomorphisms of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294070.png" />-sphere onto itself are differentiably isotopic. The isotopy classes of differentiable imbeddings of a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294071.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294072.png" /> have been studied in detail for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294073.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294074.png" />, then there exist a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294076.png" />, such that the homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294077.png" /> are in one-to-one correspondence with the imbedding classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294078.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294079.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294080.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294081.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294082.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294083.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294084.png" /> depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294085.png" /> being even or odd. Thus, the standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294086.png" />-sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294087.png" />, imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294088.png" />, can for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294089.png" /> be knotted in the differentiable sense, that is, there exist imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294090.png" /> that are not differentiably isotopic to the standard imbedding. These knots are called Haefliger knots. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294091.png" />, then the differentiable knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294092.png" /> can be knotted in the topological sense; however, there are considerably more of these than of the topological or piecewise-linear ones; for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294093.png" /> they have been completely classified. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294095.png" />, then any differentiable imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294096.png" /> is differentiably isotopic to the standard imbedding.

The fact that there exist diffeomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294097.png" /> onto itself that are not isotopic to the identity leads to the existence of non-trivial differentiable structures on spheres of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294098.png" />. Although every homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i05294099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i052940100.png" />, onto itself can can be approximated by a diffeomorphism, not all close diffeomorphisms of a sphere are differentiably isotopic, that is, a diffeomorphism of a sphere onto itself can by an infinitesimal perturbation be transformed in one not isotopic to it.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Keldysh,   "Topological imbeddings in Euclidean space"  ''Proc. Steklov Inst. Math.'' , '''81'''  (1968)  ''Trudy Mat. Inst. Steklov.'' , '''81'''</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.P. Rourke,   B.J. Sanderson,   "Introduction to piecewise-linear topology" , Springer  (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Novikov,   "Differentiable sphere bundles"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''29''' :  1  (1965)  pp. 71–96  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Chernavskii,   "Local contractibility of the group of homeomorphisms of a manifold"  ''Math. USSR Sb.'' , '''8''' :  3  (1969)  pp. 287–333</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T. Rushing,   "Topological embeddings" , Acad. Press  (1973)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.C. Siebenmann,   R.C. Kirby,   "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press  (1977)</TD></TR></table>



====Comments====
Isotopies are also very important in infinite-dimensional topology (cf. [[Infinite-dimensional space|Infinite-dimensional space]]). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i052940101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i052940102.png" /> are points in the [[Hilbert cube|Hilbert cube]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i052940103.png" />, then there exists an isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i052940104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i052940105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052940/i052940106.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. van Mill,   "Infinite-dimensional topology, prerequisites and introduction" , North-Holland  (1988)</TD></TR></table>