The Milnor Fiber as a Local Conley Index
Abstract
Let f:(Cn,0)→(C,0) be a holomorphic germ with an isolated critical point at the origin, and let φf 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 φf and that its Conley index is the suspension of the Milnor fiber, h(φf,{0}) ≃ ΣFf ≃ bigveeμ(f)Sn , so that μ(f)=rank CHn(φf,{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 μ(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 −∇re f.
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BibTeX
@misc{brasselet2026milnor,
title = {{The Milnor Fiber as a Local Conley Index}},
author = {Jean-Paul Brasselet and Nivaldo Grulha},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10896},
eprint = {2610.10896},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10896},
}
RIS
TY - UNPB
TI - The Milnor Fiber as a Local Conley Index
AU - Jean-Paul Brasselet
AU - Nivaldo Grulha
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10896
UR - https://arxiv.org/abs/2610.10896
AB - Let f:(C^n,0)→(C,0) be a holomorphic germ with an isolated critical point at the origin, and let φ_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 φ_f and that its Conley index is the suspension of the Milnor fiber, h(φ_f,{0}) ≃ ΣF_f ≃ 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 μ(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 −∇re f.
ER -