Preimage theorem

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In mathematics, particularly in differential topology, the preimage theorem is a theorem concerning the preimage of particular points in a manifold under the action of a smooth map.

[edit] 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 $df_x: T_xX \to T_yY\,\!$ is surjective. Here, $T_xX\,\!$ and $T_yY\,\!$ 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 $\{x : x \in f^{-1}(y)\}$ is a submanifold of X. Further, the codimension of this manifold in X is equal to the dimension of Y.