Milnor map

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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.