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) |
[2] | M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) |
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) |
Finite-to-one mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Finite-to-one_mapping&oldid=24440