View source for Finite-to-one mapping

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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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) {{MR|785749}} {{ZBL|0568.54001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) {{MR|0341518}} {{ZBL|0294.58004}} </TD></TR></table>

====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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) {{MR|777682}} {{ZBL|0554.58001}} </TD></TR></table>