Milnor number

In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ.
If "f" is a complex-valued holomorphic function germ then the Milnor number of "f", denoted "μ"("f"), is either a nonnegative integer, or is infinite. It can be considered both a geometric invariant and an algebraic invariant. This is why it plays an important role in algebraic geometry and singularity theory.
Algebraic definition.
Consider a holomorphic complex function germ 
and denote by formula_2 the ring of all function germs formula_3. 
Every level of a function is a complex hypersurface in formula_4, therefore we will call formula_5 a hypersurface singularity.
Assume it is an isolated singularity: in case of holomorphic mappings we say that a hypersurface singularity formula_5 is singular at formula_7 if its gradient formula_8 is zero at formula_9, a singular point is isolated if it is the only singular point in a sufficiently small neighbourhood. In particular, the multiplicity of the gradient
is finite. This number formula_11 is the Milnor number of singularity formula_12 at formula_13.
Note that the multiplicity of the gradient is finite if and only if the origin is an isolated critical point of "f".
Geometric interpretation.
Milnor originally introduced formula_14 in geometric terms in the following way. All fibers formula_15 for values formula_16 close to formula_13 are nonsingular manifolds of real dimension formula_18. Their intersection with a small open disc formula_19 centered at formula_13 is a smooth manifold formula_21 called the Milnor fiber. Up to diffeomorphism formula_21 does not depend on formula_16 or formula_24 if they are small enough. It is also diffeomorphic to the fiber of the Milnor fibration map.
The Milnor fiber formula_21 is a smooth manifold of dimension formula_18 and has the same homotopy type as a bouquet of formula_14 spheres formula_28. This is to say that its middle Betti number formula_29 is equal to the Milnor number and it has homology of a point in dimension less than formula_30. For example, a complex plane curve near every singular point formula_31 has its Milnor fiber homotopic to a wedge of formula_32 circles (Milnor number is a local property, so it can have different values at different singular points).
Thus we have equalities