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.

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

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