Preimage theorem explained

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.^{[1] [2]}

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 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.

Theorem. Let

f: X \to Y be a smooth map, and let y \in Y be a regular value of f. Then f ^{-1} (y) is a submanifold of X. If y \in 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 \ker(df _{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 (dg _{x)} = m for all x \in g ^{-1} (y). Then g ^{-1} (y) is a complex submanifold of X of complex dimension n - m.

Notes and References

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