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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i0502401.png" /> of one topological space into another for which each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i0502402.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i0502403.png" /> which is homeomorphically mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i0502404.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i0502405.png" />. This concept is applied mainly to mappings of manifolds, where one often additionally requires a local flatness condition (as for a [[Locally flat imbedding|locally flat imbedding]]). The latter condition is automatically fulfilled if the manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i0502406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i0502407.png" /> are differentiable and if the Jacobi matrix of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i0502408.png" /> has maximum rank, equal to the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i0502409.png" /> at each point. The classification of immersions of one manifold into another up to a regular homotopy can be reduced to a pure homotopic problem. A [[Homotopy|homotopy]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024010.png" /> is called regular if for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024011.png" /> it can be continued to an [[Isotopy (in topology)|isotopy (in topology)]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024013.png" /> is a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024015.png" /> is a disc of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024017.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024019.png" />, where 0 is the centre of the disc. In the differentiable case, it is sufficient to require that the Jacobi matrix has maximum rank for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024020.png" /> and depends continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024021.png" />. The differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024022.png" /> of an immersion determines a fibre-wise monomorphism of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024023.png" /> into the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024024.png" />. A regular homotopy determines a homotopy of such monomorphisms. This establishes a bijection between the classes of regular homotopies and the homotopy classes of monomorphisms of bundles.
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The problem of immersions in a Euclidean space reduces to the homotopy classification of mappings into a [[Stiefel manifold|Stiefel manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024025.png" />. For example, because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024026.png" />, there is only one immersion class of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024027.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024028.png" />, so the standard imbedding is regularly homotopic to its mirror reflection (the sphere may be regularly turned inside out. Because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024029.png" />, there is a countable number of immersion classes of a circle into the plane, and because the Stiefel fibration over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024030.png" /> is homeomorphic to the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024032.png" />, there are only two immersion classes from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024033.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024034.png" />, etc.
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A mapping  $  f:  X \rightarrow Y $
 +
of one topological space into another for which each point of  $  X $
 +
has a neighbourhood  $  U $
 +
which is homeomorphically mapped onto  $  fU $
 +
by  $  f $.
 +
This concept is applied mainly to mappings of manifolds, where one often additionally requires a local flatness condition (as for a [[Locally flat imbedding|locally flat imbedding]]). The latter condition is automatically fulfilled if the manifolds  $  X $
 +
and  $  Y $
 +
are differentiable and if the Jacobi matrix of the mapping  $  f $
 +
has maximum rank, equal to the dimension of  $  X $
 +
at each point. The classification of immersions of one manifold into another up to a regular homotopy can be reduced to a pure homotopic problem. A [[Homotopy|homotopy]]  $  f _ {t} :  X  ^ {m} \rightarrow Y  ^ {n} $
 +
is called regular if for each point  $  x \in X $
 +
it can be continued to an [[Isotopy (in topology)|isotopy (in topology)]]  $  F _ {t} :  U \times D  ^ {k} \rightarrow Y $,
 +
where  $  U $
 +
is a neighbourhood of  $  x $,
 +
$  D  ^ {k} $
 +
is a disc of dimension  $  k = n- m $
 +
and  $  F _ {t} $
 +
coincides with  $  f _ {t} $
 +
on  $  U \times 0 $,
 +
where 0 is the centre of the disc. In the differentiable case, it is sufficient to require that the Jacobi matrix has maximum rank for each  $  t $
 +
and depends continuously on  $  t $.
 +
The differential  $  D _ {f} $
 +
of an immersion determines a fibre-wise monomorphism of the tangent bundle  $  \tau X $
 +
into the tangent bundle  $  \tau Y $.
 +
A regular homotopy determines a homotopy of such monomorphisms. This establishes a bijection between the classes of regular homotopies and the homotopy classes of monomorphisms of bundles.
  
 +
The problem of immersions in a Euclidean space reduces to the homotopy classification of mappings into a [[Stiefel manifold|Stiefel manifold]]  $  V _ {n,m }  $.
 +
For example, because  $  \pi _ {2} ( V _ {3,2 }  ) = 0 $,
 +
there is only one immersion class of the sphere  $  S  ^ {2} $
 +
into  $  \mathbf R  ^ {3} $,
 +
so the standard imbedding is regularly homotopic to its mirror reflection (the sphere may be regularly turned inside out. Because  $  V _ {2,1 }  \approx S  ^ {1} $,
 +
there is a countable number of immersion classes of a circle into the plane, and because the Stiefel fibration over  $  S  ^ {2} $
 +
is homeomorphic to the projective space  $  \mathbf R P  ^ {3} $
 +
and  $  \pi _ {1} ( \mathbf R P  ^ {3} ) = \mathbf Z _ {2} $,
 +
there are only two immersion classes from  $  S  ^ {1} $
 +
into  $  S  ^ {2} $,
 +
etc.
  
 
====Comments====
 
====Comments====
For figures illustrating the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050240/i05024035.png" /> can be regularly turned inside out see [[#References|[a3]]].
+
For figures illustrating the fact that $  S  ^ {2} $
 +
can be regularly turned inside out see [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.L. Gromov, "Stable mappings of foliations into manifolds" ''Math. USSR Izv.'' , '''3''' (1969) pp. 671–694 ''Izv. Akad. Nauk SSSR'' , '''33''' (1969) pp. 707–734 {{MR|0263103}} {{ZBL|0205.53502}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V. Poénaru, "Homotopy theory and differentiable singularities" N.H. Kuiper (ed.) , ''Manifolds (Amsterdam, 1970)'' , ''Lect. notes in math.'' , '''197''' , Springer (1971) pp. 106–132 {{MR|0285026}} {{ZBL|0215.52802}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Phillips, "Turning a surface inside out" ''Scientific Amer.'' , '''May''' (1966) pp. 112–120</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.L. Gromov, "Stable mappings of foliations into manifolds" ''Math. USSR Izv.'' , '''3''' (1969) pp. 671–694 ''Izv. Akad. Nauk SSSR'' , '''33''' (1969) pp. 707–734 {{MR|0263103}} {{ZBL|0205.53502}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V. Poénaru, "Homotopy theory and differentiable singularities" N.H. Kuiper (ed.) , ''Manifolds (Amsterdam, 1970)'' , ''Lect. notes in math.'' , '''197''' , Springer (1971) pp. 106–132 {{MR|0285026}} {{ZBL|0215.52802}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Phillips, "Turning a surface inside out" ''Scientific Amer.'' , '''May''' (1966) pp. 112–120</TD></TR></table>

Latest revision as of 22:11, 5 June 2020


A mapping $ f: X \rightarrow Y $ of one topological space into another for which each point of $ X $ has a neighbourhood $ U $ which is homeomorphically mapped onto $ fU $ by $ f $. This concept is applied mainly to mappings of manifolds, where one often additionally requires a local flatness condition (as for a locally flat imbedding). The latter condition is automatically fulfilled if the manifolds $ X $ and $ Y $ are differentiable and if the Jacobi matrix of the mapping $ f $ has maximum rank, equal to the dimension of $ X $ at each point. The classification of immersions of one manifold into another up to a regular homotopy can be reduced to a pure homotopic problem. A homotopy $ f _ {t} : X ^ {m} \rightarrow Y ^ {n} $ is called regular if for each point $ x \in X $ it can be continued to an isotopy (in topology) $ F _ {t} : U \times D ^ {k} \rightarrow Y $, where $ U $ is a neighbourhood of $ x $, $ D ^ {k} $ is a disc of dimension $ k = n- m $ and $ F _ {t} $ coincides with $ f _ {t} $ on $ U \times 0 $, where 0 is the centre of the disc. In the differentiable case, it is sufficient to require that the Jacobi matrix has maximum rank for each $ t $ and depends continuously on $ t $. The differential $ D _ {f} $ of an immersion determines a fibre-wise monomorphism of the tangent bundle $ \tau X $ into the tangent bundle $ \tau Y $. A regular homotopy determines a homotopy of such monomorphisms. This establishes a bijection between the classes of regular homotopies and the homotopy classes of monomorphisms of bundles.

The problem of immersions in a Euclidean space reduces to the homotopy classification of mappings into a Stiefel manifold $ V _ {n,m } $. For example, because $ \pi _ {2} ( V _ {3,2 } ) = 0 $, there is only one immersion class of the sphere $ S ^ {2} $ into $ \mathbf R ^ {3} $, so the standard imbedding is regularly homotopic to its mirror reflection (the sphere may be regularly turned inside out. Because $ V _ {2,1 } \approx S ^ {1} $, there is a countable number of immersion classes of a circle into the plane, and because the Stiefel fibration over $ S ^ {2} $ is homeomorphic to the projective space $ \mathbf R P ^ {3} $ and $ \pi _ {1} ( \mathbf R P ^ {3} ) = \mathbf Z _ {2} $, there are only two immersion classes from $ S ^ {1} $ into $ S ^ {2} $, etc.

Comments

For figures illustrating the fact that $ S ^ {2} $ can be regularly turned inside out see [a3].

References

[a1] M.L. Gromov, "Stable mappings of foliations into manifolds" Math. USSR Izv. , 3 (1969) pp. 671–694 Izv. Akad. Nauk SSSR , 33 (1969) pp. 707–734 MR0263103 Zbl 0205.53502
[a2] V. Poénaru, "Homotopy theory and differentiable singularities" N.H. Kuiper (ed.) , Manifolds (Amsterdam, 1970) , Lect. notes in math. , 197 , Springer (1971) pp. 106–132 MR0285026 Zbl 0215.52802
[a3] A. Phillips, "Turning a surface inside out" Scientific Amer. , May (1966) pp. 112–120
How to Cite This Entry:
Immersion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Immersion&oldid=47317
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article