In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map.
Statement of Theorem
Definition. Let f : X → Y {\displaystyle f:X\to Y} be a smooth map between manifolds. We say that a point y ∈ Y {\displaystyle y\in Y} is a regular value of f {\displaystyle f} if for all x ∈ f − 1 ( y ) {\displaystyle x\in f^{-1}(y)} the map d f x : T x X → T y Y {\displaystyle df_{x}:T_{x}X\to T_{y}Y} is surjective. Here, T x X {\displaystyle T_{x}X} and T y Y {\displaystyle T_{y}Y} are the tangent spaces of X {\displaystyle X} and Y {\displaystyle Y} at the points x {\displaystyle x} and y . {\displaystyle y.}
Theorem. Let f : X → Y {\displaystyle f:X\to Y} be a smooth map, and let y ∈ Y {\displaystyle y\in Y} be a regular value of f . {\displaystyle f.} Then f − 1 ( y ) {\displaystyle f^{-1}(y)} is a submanifold of X . {\displaystyle X.} If y ∈ im ( f ) , {\displaystyle y\in {\text{im}}(f),} then the codimension of f − 1 ( y ) {\displaystyle f^{-1}(y)} is equal to the dimension of Y . {\displaystyle Y.} Also, the tangent space of f − 1 ( y ) {\displaystyle f^{-1}(y)} at x {\displaystyle x} is equal to ker ( d f x ) . {\displaystyle \ker(df_{x}).}
There is also a complex version of this theorem:3
Theorem. Let X n {\displaystyle X^{n}} and Y m {\displaystyle Y^{m}} be two complex manifolds of complex dimensions n > m . {\displaystyle n>m.} Let g : X → Y {\displaystyle g:X\to Y} be a holomorphic map and let y ∈ im ( g ) {\displaystyle y\in {\text{im}}(g)} be such that rank ( d g x ) = m {\displaystyle {\text{rank}}(dg_{x})=m} for all x ∈ g − 1 ( y ) . {\displaystyle x\in g^{-1}(y).} Then g − 1 ( y ) {\displaystyle g^{-1}(y)} is a complex submanifold of X {\displaystyle X} of complex dimension n − m . {\displaystyle n-m.}
See also
- Fiber (mathematics) – Set of all points in a function's domain that all map to some single given point
- Level set – Subset of a function's domain on which its value is equal
References
Tu, Loring W. (2010), "9.3 The Regular Level Set Theorem", An Introduction to Manifolds, Springer, pp. 105–106, ISBN 9781441974006. 9781441974006 ↩
Banyaga, Augustin (2004), "Corollary 5.9 (The Preimage Theorem)", Lectures on Morse Homology, Texts in the Mathematical Sciences, vol. 29, Springer, p. 130, ISBN 9781402026959. 9781402026959 ↩
Ferrari, Michele (2013), "Theorem 2.5", Complex manifolds - Lecture notes based on the course by Lambertus Van Geemen (PDF). http://www.mat.unimi.it/users/geemen/Ferrari_complexmanifolds.pdf ↩