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