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
f : ( C n , 0 ) → ( C , 0 ) {\displaystyle f:(\mathbb {C} ^{n},0)\to (\mathbb {C} ,0)\ }
and denote by O n {\displaystyle {\mathcal {O}}_{n}} the ring of all function germs ( C n , 0 ) → ( C , 0 ) {\displaystyle (\mathbb {C} ^{n},0)\to (\mathbb {C} ,0)} . Every level of a function is a complex hypersurface in C n {\displaystyle \mathbb {C} ^{n}} , therefore f {\displaystyle f} is dubbed a hypersurface singularity. Assume it is an isolated singularity: in the case of holomorphic mappings it is said that a hypersurface singularity f {\displaystyle f} is singular at 0 ∈ C n {\displaystyle 0\in \mathbb {C} ^{n}} if its gradient ∇ f {\displaystyle \nabla f} is zero at 0 {\displaystyle 0} , and it is said that 0 {\displaystyle 0} is an isolated singular point if it is the only singular point in a sufficiently small neighbourhood of 0 {\displaystyle 0} . In particular, the multiplicity of the gradient
μ ( f ) = dim C O n / ∇ f {\displaystyle \mu (f)=\dim _{\mathbb {C} }{\mathcal {O}}_{n}/\nabla f}
is finite by an application of Rückert's Nullstellensatz. This number μ ( f ) {\displaystyle \mu (f)} is the Milnor number of singularity f {\displaystyle f} at 0 {\displaystyle 0} . 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 μ ( f ) {\displaystyle \mu (f)} in geometric terms in the following way. All fibers f − 1 ( c ) {\displaystyle f^{-1}(c)} for values c {\displaystyle c} close to 0 {\displaystyle 0} are nonsingular manifolds of real dimension 2 ( n − 1 ) {\displaystyle 2(n-1)} . Their intersection with a small open disc D ϵ {\displaystyle D_{\epsilon }} centered at 0 {\displaystyle 0} is a smooth manifold F {\displaystyle F} called the Milnor fiber. Up to diffeomorphism F {\displaystyle F} does not depend on c {\displaystyle c} or ϵ {\displaystyle \epsilon } if they are small enough. It is also diffeomorphic to the fiber of the Milnor fibration map. The Milnor fiber F {\displaystyle F} is a smooth manifold of dimension 2 ( n − 1 ) {\displaystyle 2(n-1)} and has the same homotopy type as a bouquet of μ ( f ) {\displaystyle \mu (f)} spheres S n − 1 {\displaystyle S^{n-1}} . This is to say that its middle Betti number b n − 1 ( F ) {\displaystyle b_{n-1}(F)} is equal to the Milnor number and it has homology of a point in dimension less than n − 1 {\displaystyle n-1} . For example, a complex plane curve near every singular point z 0 {\displaystyle z_{0}} has its Milnor fiber homotopic to a wedge of μ z 0 ( f ) {\displaystyle \mu _{z_{0}}(f)} circles (Milnor number is a local property, so it can have different values at different singular points). Thus the following equalities hold:
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