Milnor map

Definition

Let $f$ be a complex polynomial in $(n+1)$ variables. $f$ can be viewed as a map $\mathbb {R} ^{2n+2}\to \mathbb {R} ^{2}$. Let $V_{f}$ denote the zero set of $f$. Then, on the complement of $V_{f}$, we can define a map $f/|f|$ from $\mathbb {R} ^{2n+2}\setminus V_{f}$ 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 $S^{1}$.

Facts=

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.