View source for Immersion

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.

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.



====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]]].

====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>