# Immersion

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=24473