The Milnor Fiber as a Local Conley Index

Let $f:({\text{C}}^n,0)\to (\text{C},0)$ be a holomorphic germ with an isolated critical point at the origin, and let ${\phi }_f$ be the negative gradient flow of $\text{re}f$ with respect to a Riemannian metric. We show that the origin is an isolated invariant set of ${\phi }_f$ and that its Conley index is the suspension of the Milnor fiber, $$h({\phi }_f,\{0\})\simeq \mathrm{\Sigma }{F}_f\simeq \underset{\mu (f)}{\bigvee }{S}^n,$$ so that $\mu (f)=\text{rank}{\text{CH}}_n({\phi }_f,\{0\};\mathbb{Z})$: the Milnor number is the rank of the homological Conley index of the flow in degree $n$. The proof uses continuation to a holomorphic morsification of $f$. Since $\text{im}f$ is a first integral of the flow, after a rotation of the target through an angle avoiding finitely many values there are no connecting orbits between the $\mu (f)$ nondegenerate rest points that result, and the sum property of the Conley index applies. McCord’s theorem then recovers the Poincaré–Hopf index of $-\mathrm{\nabla }\text{re}f$.