# Milnor map

Let $f$ be a complex polynomial in $(n+1)$ variables. $f$ can be viewed as a map $\R^{2n+2} \to \R^2$. Let $V_f$ denote the zero set of $f$. Then, on the complement of $V_f$, we can define a map $f/|f|$ from $\R^{2n+2} \setminus V_f$ to $S^1$.
The Milnor map of $f$ at radius $r$ is the restriction of this map to the sphere of radius $r$, centered at the origin, to $S^1$.
By the Milnor fibration theorem, the Milnor map is a fibration whenever the origin is an isolated singular point of $V_f$. Under such circumstances, it is termed the Milnor fibration.