arXiv subjects

Sakumi Sugawara

Publications and source records attributed to Sakumi Sugawara.

10 recordsLinked to original sources

arXiv2026

Topological line arrangements and their topological invariants

A topological line arrangement is an arrangement of embedded spheres in the complex projective plane that topologically generalizes a complex line arrangement. In this paper, we establish foundational results on the topology of the complement of topological line arrangements. First, we prove that the cohomology ring of the complement is isomorphic to the Orlik-Solomon algebra, as for classical complex line arrangements. We then study the homotopy type of the complement. We prove that the complement of a symplectic line arrangement has the homotopy type of a minimal CW complex. In contrast, every combinatorial type realizable by a topological line arrangement admits a realization with a non-minimal complement. Moreover, every such combinatorial type admits infinitely many realizations whose complements are pairwise non-homotopy equivalent.

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arXiv2026

Even torsions in the homology group of the Milnor fiber boundary of hyperplane arrangements in $\mathbb{C}^3$

We study the homology group of the Milnor fiber boundary of a hyperplane arrangement in $\mathbb{C}^{3}$. By the work of Némethi--Szilárd, the homeomorphism type of the Milnor fiber boundary is combinatorially determined, and an explicit formula for the first Betti number is known. However, the torsion part of the first homology group is poorly understood. In this paper, under some conditions, we prove that the number of even-order torsion summands of the first homology group is greater than or equal to the Euler characteristic of the projectivized complement.

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arXiv2026

The cohomology ring of the boundary manifold of a combinatorial line arrangement

We compute the integral cohomology rings of the associated $3$-manifold called the \textit{boundary manifold} for a combinatorial line arrangement, and prove that it is isomorphic to the double of the Orlik-Solomon algebra. This generalizes the Cohen-Suciu doubling formula for complex realizable case to arbitrary combinatorial line arrangements, including non-realizable ones. To handle the non-realizable case without geometric realization, we construct explicit homology cycles and compute intersection products, following the method of Doig--Horn for graph manifolds. As an application, we derive several results on the resonance variety of the boundary manifold.

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arXiv2026

Handle decompositions and Kirby diagrams for the complement of plane algebraic curves

The complement of plane algebraic curves are well studied from topological and algebro-geometric viewpoints. In this paper, we will describe the explicit handle decompositions and the Kirby diagrams for the complement of plane algebraic curves. The method is based on the notion of braid monodromy. We refined this technique to obtain handle decompositions and Kirby diagrams.

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arXiv2025

First homology groups of the Milnor fiber boundary for generic hyperplane arrangements in $\mathbb{C}^{3}$

We study the Milnor fiber boundary for hyperplane arrangements in $\mathbb{C}^3$. This is one of the examples of non-isolated surface singularities, which are studied by Némethi--Szilárd. In this paper, we compute the first homology group of the Milnor fiber boundary for a generic arrangement, which gives an affirmative answer to the conjecture of Suciu. Also, we give an example of an arrangement with $n$ hyperplanes, whose torsion part in the Milnor fiber boundary homology contains a direct summand other than $\mathbb{Z}_{n}$, for certain value of $n$.

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arXiv2024

Homogeneous quandles with abelian inner automorphism groups

In this paper, we give a characterization of homogeneous quandles with abelian inner automorphism groups. In particular, we show that such a quandle is expressed as an abelian extension of a trivial quandle. Our construction is a generalization of the recent work by Furuki and Tamaru, which gives a construction of disconnected flat quandles.

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arXiv2024

Betti numbers and torsions in homology groups of double coverings

Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial $2$-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent.

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arXiv2024

Divides with cusps and symmetric links

A Divide with cusps is the image of a proper generic immersion from finite intervals and circles into a $2$-disk which allows to have cusps. A divide with cusps is the generalization of the notion of the divide which is introduced by A'Campo. From a divide with cusps, we can define the associated link in $S^3$. In this paper, we give the characterization of the link in $S^3$ which can be described as the associated link of a divide with cusps. In particular, we prove that every strongly invertible link and $2$-periodic link can be described as the link of a divide with cusps.

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arXiv2023

$\mathbb{Z}$-local system cohomology of hyperplane arrangements and a Cohen-Dimca-Orlik type theorem

Local system cohomology groups of the complements of hyperplane arrangements have played an important role in the theory of hypergeometric integrals, the topology of Milnor fibers and covering spaces. One of the important theorems is the vanishing theorem for generic $\mathbb{C}$-local systems which goes back to Aomoto's work. Later, Cohen, Dimca, and Orlik proved a stronger version of the vanishing theorem. In this paper, we prove a Cohen-Dimca-Orlik type theorem for $\mathbb{Z}$-local systems.

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arXiv2021

Divides with cusps and Kirby diagrams for line arrangements

The complement of a complexified real line arrangement is an affine surface. It is classically known that such a space has a handle decomposition up to $2$-handles. We will describe the handle decomposition induced from Lefschetz hyperplane section theorem for such a space. To describe the Kirby diagram, we introduce the notion of the divide with cusps which is a generalization of the divide introduced by A'Campo.

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