Preimage theorem

 ( Theorems In Differential Topology ) 

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

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Statement of Theorem Definition. Let $f\; :\; X\; \backslash to\; Y$ be a smooth map between manifolds. We say that a point $y\; \backslash in\; Y$ is a regular value of $f$ if for all $x\; \backslash in\; f^\{-1\}(y)$ the map $d\; f\_x:\; T\_x\; X\; \backslash to\; T\_y\; Y$ is surjective map. Here, $T\_x\; X$ and $T\_y\; Y$ are the of $X$ and $Y$ at the points $x$ and $y.$

Theorem. Let $f:\; X\; \backslash to\; Y$ be a smooth map, and let $y\; \backslash in\; Y$ be a regular value of $f.$ Then $f^\{-1\}(y)$ is a submanifold of $X.$ If $y\; \backslash in\; \backslash text\{im\}(f),$ 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 $\backslash ker(df\_x).$

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 $g\; :\; X\; \backslash to\; Y$ be a holomorphic map and let $y\; \backslash in\; \backslash text\{im\}(g)$ be such that $\backslash text\{rank\}(dg\_x)\; =\; m$ for all $x\; \backslash in\; g^\{-1\}(y).$ Then $g^\{-1\}(y)$ is a complex submanifold of $X$ of complex dimension $n\; -\; m.$

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See also

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