Finite-to-one mapping

A mapping such that the number of points in the pre-image of every point is finite. If is the same for all , is said to be an -to-one mapping.

In the differentiable case, the concept of a finite-to-one mapping corresponds to that of a finite mapping. A differentiable mapping of differentiable manifolds is said to be finite at a point if the dimension of the local ring of at is finite. All mappings of this sort are finite-to-one mappings on compact subsets of ; moreover, there exists an open neighbourhood of such that consists of a single point. The number measures the multiplicity of as a root of the equation ; there exists a neighbourhood of such that has at most points for every sufficiently close to .

If , the finite mappings form a generic set in the space ; moreover, the set of non-finite mappings has infinite codimension in that space (Tougeron's theorem).

References

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) MR785749 Zbl 0568.54001
[2]	M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) MR0341518 Zbl 0294.58004

Comments

Let be a mapping of differentiable manifolds. For let denote the ring of germs of smooth functions at . This is a local ring with maximal ideal consisting of all germs vanishing at . If , then by pullback, induces a ring homomorphism . The local ring of the mapping is now defined as the quotient ring .

If are germs of stable mappings then and are equivalent if and only if and are isomorphic as rings (Mather's theorem). Here equivalence of germs of mappings means that there exist germs of diffeomorphisms and such that (near ).

References

[a1]	V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 1 , Birkhäuser (1985) (Translated from Russian) MR777682 Zbl 0554.58001