Combinatorial Determination of Arrangement Milnor-Fiber Betti Numbers

Definitions and mathematical statement

The supplied question asks whether the Betti numbers of an arrangement Milnor fiber are determined by the arrangement's combinatorial data. It does not define that data, specify centrality or multiplicities, or state a coefficient convention for Betti numbers. The following is a precise conventional formulation, conditional on interpreting the source as referring to finite, nonempty, central, reduced complex hyperplane arrangements and their intersection lattices.

Let $\mathcal A$ be a finite nonempty set of distinct complex linear hyperplanes in ${\mathbb{C}}^{\ell}$, where $\ell\geq 1$. For each $H\in\mathcal A$, choose a nonzero complex linear form $\alpha_H$ with kernel $H$, and put \[ Q_{\mathcal A}(z)=\prod_{H\in\mathcal A}\alpha_H(z),\qquad F_{\mathcal A}=\{z\in{\mathbb{C}}^{\ell}:Q_{\mathcal A}(z)=1\}. \] Here $F_{\mathcal A}$ is the usual global Milnor fiber of the reduced homogeneous arrangement polynomial. Define \[ b_i(F_{\mathcal A})=\dim_{{\mathbb{Q}}}H_i(F_{\mathcal A};{\mathbb{Q}}),\qquad i\in{\mathbb{Z}}_{\geq 0}. \] Let $L(\mathcal A)$ be the poset of intersections of subcollections of $\mathcal A$, ordered by reverse inclusion, with the empty intersection interpreted as ${\mathbb{C}}^{\ell}$. Under the conventional interpretation $d_{\mathcal A}=[L(\mathcal A)]$, the question is whether, for every pair of such arrangements $\mathcal A\subset{\mathbb{C}}^{\ell}$ and $\mathcal B\subset{\mathbb{C}}^{m}$, and every choice of their defining linear forms, \[ L(\mathcal A)\cong L(\mathcal B) \quad\Longrightarrow\quad b_i(F_{\mathcal A})=b_i(F_{\mathcal B}) \quad\text{for every }i\in{\mathbb{Z}}_{\geq 0}. \] No essentiality hypothesis is imposed in this conditional formulation. The identification of $d_{\mathcal A}$ with the intersection lattice, the central reduced setting, and the rational coefficient convention require confirmation against the intended source. If affine arrangements, multiarrangements, another Milnor-fiber construction, or richer combinatorial data were intended, the formulation would need revision.

Short English statement

Under the explicitly stated central, reduced arrangement and intersection-lattice conventions, the question asks whether two complex hyperplane arrangements with isomorphic intersection lattices must have Milnor fibers with the same Betti numbers in every degree. The supplied record labels this question open; that label is supplied curation, not an independently verified status assessment. The intended conventions require human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30003693$, source problem code catalogue entry, title: Combinatorial Determination of Arrangement Milnor-Fiber Betti Numbers. The supplied record belongs to the Oberwolfach Reports Open Problems collection and records proposed year 2018; no individual proposer is supplied.
• Supplied report citation: Topology of Arrangements and Representation Stability (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/2, pp. 43--123. Supplied source URL and literature-source link, labeled "Oberwolfach report": https://doi.org/10.4171/owr/2018/2. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. This citation was supplied and has not been independently checked.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the Milnor-fiber Betti-number question open and states that a separate same-combinatorics arrangement-group problem was resolved. This is supplied dated curation, not an independently verified literature assessment; the source itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with an accompanying assertion that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This is a registry assertion and does not independently verify the interpretive conventions in this draft.