Preimage theorem

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.

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 $$y \in \text{im}(f),$$ 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 $$ \ker(df_x).$$

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