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.^[1]^[2]

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 $d f_{x} : T_{x} X \to T_{y} Y$ is surjective. Here, $T_{x} X$ and $T_{y} Y$ are the tangent spaces of $X$ and $Y$ at the points $x$ and $y .$ A submanifold $S \subseteq X$ is called smoothly embedded if the inclusion map $ι : S ↪ X$ is a smooth embedding, and properly embedded if $ι$ 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 $T_{x} S = \ker (d f_{x})$ , where $\ker (d f_{x})$ is the kernel of the differential of $f$ at $x$ .

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

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 $y \in im (g)$ be such that $rank (d g_{x}) = m$ for all $x \in g^{- 1} (y) .$ Then $g^{- 1} (y)$ is a complex submanifold of $X$ of complex dimension $n - m .$

References

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

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[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).

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Statement of Theorem

See also

References