Combinatorial Determination of Milnor-Fiber Monodromy

Definitions and mathematical statement

The supplied question concerns an arrangement $A$, its arrangement intersection data $d_A$, an associated Milnor fiber $F_A$, and the monodromy homomorphism on $H^1(F_A)$. The record does not define the class of arrangements, the data $d_A$, the construction of $F_A$, or the cohomology coefficients.

To describe a possible precise formulation without fixing these missing choices, let $\mathcal C$ denote the intended class of arrangements and let $R$ denote a fixed coefficient ring. Once a Milnor fibration and a geometric monodromy homeomorphism $h_A:F_A\to F_A$ have been specified, write \[ G_A=H^1(F_A;R),\qquad T_A=h_A^*:G_A\longrightarrow G_A, \] where $H^1$ denotes first singular cohomology. Two pairs $(G_A,T_A)$ and $(G_B,T_B)$ are isomorphic if there is an $R$-module isomorphism $\phi:G_A\to G_B$ satisfying $\phi\circ T_A=T_B\circ\phi$.

Under the interpretation that "determined" means determined up to this isomorphism, the question is whether \[ \forall A,B\in\mathcal C,\qquad d_A\cong d_B \ \Longrightarrow\ \exists\,\phi:G_A\xrightarrow{\sim}G_B \quad\text{such that}\quad \phi\circ T_A=T_B\circ\phi. \] Here $d_A\cong d_B$ must mean the intended equivalence of arrangement intersection data. This is a conditional formulation: the record does not specify that equivalence, the class $\mathcal C$, the coefficient ring $R$, the Milnor-fiber construction and monodromy convention, or whether isomorphism of the resulting pairs is the intended meaning of determination. Those choices require source clarification before this can serve as a fully specified problem statement.

Short English statement

Does the intersection data of an arrangement determine the monodromy action on the first cohomology of its Milnor fiber? A possible interpretation asks whether arrangements with equivalent intersection data have isomorphic first cohomology modules with intertwining monodromy actions. The arrangement class, coefficients, fiber construction, and intended equivalences remain unspecified in the supplied record.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003694$, source problem code catalogue entry, title Combinatorial Determination of Milnor-Fiber Monodromy. The supplied record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is named.
• 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 record assigns proposed year 2018; the relationship between that year and the citation's 2019 date is not explained.
• Supplied literature assessment dated 2026-08-21: the report explicitly calls open the question whether Milnor-fiber first-cohomology monodromy is determined by arrangement intersection data, and no resolution was located in that curation. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies status open and statement status corrected_verified, with a note that mathematical content was preserved while notation, typography, or omitted definitions were repaired. These are supplied registry labels, not independent mathematical or source verification of this draft.