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.¹²

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 ${\displaystyle n-m.}$