preimage theorem

{{short description|On the preimage of points in a manifold under the action of a smooth map}}

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.{{citation|title=An Introduction to Manifolds|first=Loring W.|last=Tu|publisher=Springer|year=2010|isbn=9781441974006|contribution=9.3 The Regular Level Set Theorem|pages=105–106|url=https://books.google.com/books?id=xQsTJJGsgs4C&pg=PA105}}.{{citation|title=Lectures on Morse Homology|volume=29|series=Texts in the Mathematical Sciences|first=Augustin|last=Banyaga|publisher=Springer|year=2004|isbn=9781402026959|page=130|url=https://books.google.com/books?id=AX-_sbMjOK4C&pg=PA130|contribution=Corollary 5.9 (The Preimage Theorem)}}.

Statement of Theorem

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

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

There is also a complex version of this theorem:{{citation|title=Complex manifolds - Lecture notes based on the course by Lambertus Van Geemen|first=Michele|last=Ferrari|year=2013|url=http://www.mat.unimi.it/users/geemen/Ferrari_complexmanifolds.pdf|contribution=Theorem 2.5}}.

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 $n - m.$

References

Category:Theorems in differential topology

preimage theorem

Statement of Theorem

See also

References