The Milnor fiber boundary of an arrangement determines its combinatorics

TL;DR

The paper resolves the converse of a known result by proving that the boundary of the Milnor fiber $\partial F_A$ of a complex line arrangement determines its intersection poset $P_A$ for non-exceptional cases, via an explicit algorithm that reconstructs $P_A$ from the plumbing graph in normal form $G_{\mathrm{Neu}}$. It leverages the Nemethi–Szilárd framework to compute a decorated plumbing graph $G_{\mathrm{NSz}}$ from the arrangement, and then applies Neumann normalization to obtain a near-minimal graph $G$ that encodes the combinatorics. The work classifies pencils, near-pencils, and double-connected pencils as exceptional families and shows that non-exceptional arrangements are recoverable from $\partial F_A$ through the structure of $G_{\mathrm{Neu}}$, thereby linking 3-manifold topology to combinatorial data of line arrangements. This advances understanding of which topological invariants of $U_A$ and $F_A$ capture the arrangement's combinatorics and provides a concrete pipeline for recovering $P_A$ from geometric-topological data.

Abstract

The boundary of the Milnor fiber associated with a complex line arrangement is a three dimensional plumbed manifold, and it is a combinatorial invariant. We prove the reverse implication, which was conjectured Némethi and Szilárd. That is, this boundary of the Milnor fiber determines the combinatorics of the arrangement. Furthermore, we give an explicit method which constructs the poset associated with the arrangement, given a plumbing graph in normal form for the boundary.

Figures (10)

• Figure 3.6: The graph $G_{\mathrm{Neu}}$ in the case of pencils and near-pencils.
• Figure 4.1: Part of the graph $\Gamma_\mathscr{C}$ with decorations.
• Figure 4.2: The string $\mathop{\mathrm{Str}}\nolimits^\circleddash\left(1,n_j;d\,\middle|\,0,0;1\right)$ in $G_{\mathrm{NSz}}$.
• Figure 5.3: We have $d-1$ bamboos of the above form, for $i=2,\ldots,d$, and $j = d+i$. The first string shows the case when $d$ is odd, whereas the second shows the case when $d$ is even. All edges in these graphs are negative.
• Figure 5.3: To the left, the vertices with Euler number $\pm 1$ are joined by $d-1$ bamboos, each having one vertex with Euler number $0$, and one negative edge. Applying one zero-chain absorption, we get a graph with $d-2$ bamboos joining a $0$-vertex with itself. Now, apply an oriented handle absorption, R5, to each of these loops, ending with a single vertex with Euler number $0$ and genus $d-1$.

Theorems & Definitions (22)

• Theorem A
• Remark 1.2
• Lemma 2.5
• proof
• Remark 3.3
• Theorem 3.6
• proof
• Remark 3.7
• Definition 4.3
• Lemma 4.4