Immersion

A mapping of one topological space into another for which each point of has a neighbourhood which is homeomorphically mapped onto by . 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 and are differentiable and if the Jacobi matrix of the mapping has maximum rank, equal to the dimension of 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 is called regular if for each point it can be continued to an isotopy (in topology) , where is a neighbourhood of , is a disc of dimension and coincides with on , 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 and depends continuously on . The differential of an immersion determines a fibre-wise monomorphism of the tangent bundle into the tangent bundle . 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 . For example, because , there is only one immersion class of the sphere into , so the standard imbedding is regularly homotopic to its mirror reflection (the sphere may be regularly turned inside out. Because , there is a countable number of immersion classes of a circle into the plane, and because the Stiefel fibration over is homeomorphic to the projective space and , there are only two immersion classes from into , etc.

Comments

For figures illustrating the fact that 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