Preimage theorem

In mathematics, particularly in the field of differential topology, the preimage theorem, or regular value theorem, is a variation of the implicit function theorem concerning the preimage of particular points in a smooth manifold under the action of a smooth map.

Statement of Theorem

Definition. Let $X$ and $Y$ be smooth manifolds, and let $f:X\to Y$ be smooth. We say that a point $y\in Y$ is a regular value of $f$ if for all $x\in f^{-1}(y)$ the differential $df_{x}:T_{x}X\to T_{y}Y$ is surjective. Here, $T_{x}X$ and $T_{y}Y$ are the tangent spaces of $X$ and $Y$ at the points $x$ and $y.$ A submanifold $S\subseteq X$ is called smoothly embedded if the inclusion map $\iota :S\hookrightarrow X$ is a smooth embedding, and properly embedded if $\iota$ is a proper map.

Theorem. Let $X$ and $Y$ be smooth manifolds, and let $f:X\to Y$ be smooth. If $y\in Y$ is a regular value of $f$, then $S=f^{-1}(y)$ is a smoothly and properly embedded submanifold of $X.$ In addition, the codimension of $S$ is equal to the dimension of $Y$, and if $x\in S$, then $T_{x}S=\ker(df_{x})$, where $\ker(df_{x})$ is the kernel of the differential of $f$ at $x$.

There is also a complex version of this theorem:

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

See also

• Fiber (mathematics) – Class of points in a function's domain defined by having the same image by that map
• Level set – Subset of a function's domain on which its value is equal