Finite-to-one mapping

A mapping $ f: X \rightarrow Y $ such that the number $ n _ {y} $ of points in the pre-image $ f ^ { - 1 } y $ of every point $ y \in Y $ is finite. If $ n _ {y} = n $ is the same for all $ y $, $ f $ is said to be an $ n $- 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 $ f: X \rightarrow Y $ of differentiable manifolds is said to be finite at a point $ x \in X $ if the dimension of the local ring $ R _ {f} ( x) $ of $ f $ at $ x $ is finite. All mappings of this sort are finite-to-one mappings on compact subsets of $ X $; moreover, there exists an open neighbourhood $ U $ of $ x $ such that $ f ^ { - 1 } ( f ( x)) \cap U $ consists of a single point. The number $ k = \mathop{\rm dim} R _ {f} ( x) $ measures the multiplicity of $ x $ as a root of the equation $ f( y) = x $; there exists a neighbourhood $ V $ of $ x $ such that $ f ^ { - 1 } ( y) \cap V $ has at most $ k $ points for every $ y $ sufficiently close to $ x $.

If $ \mathop{\rm dim} X \leq \mathop{\rm dim} Y $, the finite mappings form a generic set in the space $ C ^ \infty ( X, Y) $; 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 $ f: X \rightarrow Y $ be a mapping of differentiable manifolds. For $ x \in X $ let $ C _ {x} ^ \infty $ denote the ring of germs of smooth functions $ X \rightarrow \mathbf R $ at $ x $. This is a local ring with maximal ideal $ \mathfrak m _ {x} $ consisting of all germs vanishing at $ x $. If $ y = f( x) $, then by pullback, $ f $ induces a ring homomorphism $ f ^ { * } : C _ {x} ^ \infty \rightarrow C _ {y} ^ \infty $. The local ring of the mapping $ f $ is now defined as the quotient ring $ R _ {f} ( x) = C _ {x} ^ \infty / C _ {x} ^ \infty f ^ {*} \mathfrak m _ {y} $.

If $ f , g : ( X , x ) \rightarrow ( Y , y ) $ are germs of stable mappings then $ f $ and $ g $ are equivalent if and only if $ R _ {f} ( x) $ and $ R _ {g} ( x) $ are isomorphic as rings (Mather's theorem). Here equivalence of germs of mappings $ f , g $ means that there exist germs of diffeomorphisms $ h: ( X , x ) \rightarrow ( X , x ) $ and $ k : ( Y , y ) \rightarrow ( Y , y ) $ such that $ g = k f h ^ {-} 1 $( near $ x $).

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