Preimage theorem

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.[1][2]

Statement of Theorem

Definition. Let ${\displaystyle f:X\to Y}$ be a smooth map between manifolds. We say that a point ${\displaystyle y\in Y}$ is a regular value of ${\displaystyle f}$ if for all ${\displaystyle x\in f^{-1}(y)}$ the map ${\displaystyle df_{x}:T_{x}X\to T_{y}Y}$ is surjective. Here, ${\displaystyle T_{x}X}$ and ${\displaystyle T_{y}Y}$ are the tangent spaces of ${\displaystyle X}$ and ${\displaystyle Y}$ at the points ${\displaystyle x}$ and ${\displaystyle y.}$

Theorem. Let ${\displaystyle f:X\to Y}$ be a smooth map, and let ${\displaystyle y\in Y}$ be a regular value of ${\displaystyle f.}$ Then ${\displaystyle f^{-1}(y)}$ is a submanifold of ${\displaystyle X.}$ If ${\displaystyle y\in {\text{im}}(f),}$ then the codimension of ${\displaystyle f^{-1}(y)}$ is equal to the dimension of ${\displaystyle Y.}$ Also, the tangent space of ${\displaystyle f^{-1}(y)}$ at ${\displaystyle x}$ is equal to ${\displaystyle \ker(df_{x}).}$

There is also a complex version of this theorem:[3]

Theorem. Let ${\displaystyle X^{n}}$ and ${\displaystyle Y^{m}}$ be two complex manifolds of complex dimensions ${\displaystyle n>m.}$ Let ${\displaystyle g:X\to Y}$ be a holomorphic map and let ${\displaystyle y\in {\text{im}}(g)}$ be such that ${\displaystyle {\text{rank}}(dg_{x})=m}$ for all ${\displaystyle x\in g^{-1}(y).}$ Then ${\displaystyle g^{-1}(y)}$ is a complex submanifold of ${\displaystyle X}$ of complex dimension ${\displaystyle n-m.}$

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

1. Tu, Loring W. (2010), "9.3 The Regular Level Set Theorem", An Introduction to Manifolds, Springer, pp. 105–106, ISBN 9781441974006.
2. 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.
3. Ferrari, Michele (2013), "Theorem 2.5", Complex manifolds - Lecture notes based on the course by Lambertus Van Geemen (PDF).