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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]].
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| ==Statement of Theorem==
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| ''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 f'' if for all <math>x \in f^{-1}(y)</math> the map <math>df_x: T_xX \to T_yY\,\!</math> is [[surjective map|surjective]]. Here, <math>T_xX\,\!</math> and <math>T_yY\,\!</math> are the [[tangent space]]s of X and Y at the points x and y.
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| ''Theorem.'' Let <math>f: X \to Y\,\!</math> be a smooth map, and let <math>y \in Y</math> be a regular value of ''f''. Then <math>f^{-1}(y) = \{x \in X : f(x) =y \}</math> is a submanifold of X. Further, if <math>y</math> is in the image of ''f'', the [[codimension]] of this manifold in X is equal to the dimension of Y, and the [[tangent space]] of <math>f^{-1}(y)</math> at a point <math>x</math> is <math>Ker(df_x)</math>.
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| {{topology-stub}}
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| [[Category:Theorems in differential topology]]
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Latest revision as of 09:42, 14 July 2014
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