{{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.<ref>{{citation|title=An Introduction to Manifolds|title-link=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|contribution-url=https://books.google.com/books?id=xQsTJJGsgs4C&pg=PA105}}.</ref><ref>{{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)}}.</ref>
==Statement of Theorem==
''Definition.'' Let <math>f : X \to Y</math> be a smooth map between manifolds. We say that a point <math>y \in Y</math> is a ''regular value of'' <math>f</math> if for all <math>x \in f^{-1}(y)</math> the map <math>d f_x: T_x X \to T_y Y</math> is surjective. Here, <math>T_x X</math> and <math>T_y Y</math> are the tangent spaces of <math>X</math> and <math>Y</math> at the points <math>x</math> and <math>y.</math>
''Theorem.'' Let <math>f: X \to Y</math> be a smooth map, and let <math>y \in Y</math> be a regular value of <math>f.</math> Then <math>f^{-1}(y)</math> is a submanifold of <math>X.</math> If <math>y \in \text{im}(f),</math> then the codimension of <math>f^{-1}(y)</math> is equal to the dimension of <math>Y.</math> Also, the tangent space of <math>f^{-1}(y)</math> at <math>x</math> is equal to <math> \ker(df_x).</math>
There is also a complex version of this theorem:<ref>{{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}}.</ref>
''Theorem.'' Let <math>X^n</math> and <math>Y^m</math> be two complex manifolds of complex dimensions <math>n > m.</math> Let <math>g : X \to Y</math> be a holomorphic map and let <math>y \in \text{im}(g)</math> be such that <math>\text{rank}(dg_x) = m</math> for all <math>x \in g^{-1}(y).</math> Then <math>g^{-1}(y)</math> is a complex submanifold of <math>X</math> of complex dimension <math>n - m.</math>
==See also==
* {{annotated link|Fiber (mathematics)}} * {{annotated link|Level set}}
==References== {{reflist}}
{{Manifolds}}
{{topology-stub}} Category:Theorems in differential topology