The Milnor Fiber as a Local Conley Index

Abstract

Let $f:(\C^n,0)\to(\C,0)$ be a holomorphic germ with an isolated critical point at the origin, and let ${\phi }_f$ be the negative gradient flow of $\re f$ 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, \[ h(φ_f,\{0\})\;\simeq\;ΣF_f\;\simeq\;\bigvee_{μ(f)}S^n , \] so that $μ(f)=\rank \CH_n(φ_f,\{0\};\Z)$: 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 $\im f$ 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 $-\nabla\re f$.