Not all free arrangements are $K(π,1)$

Freeness and near freeness are combinatorial for line arrangements in small degrees

We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane and that the property of being nearly free is combinatorial for line arrangements of up to 12 lines in the complex projective plane.

Line arrangements and direct sums of free groups

We show that if the fundamental groups of the complements of two line arrangements in the complex projective plane are isomorphic to the same direct sum of free groups, then the complements of the arrangements are homotopy equivalent. For any such arrangement, we construct another arrangement that is complexified-real, the intersection lattices of the arrangements are isomorphic, and the complements of the arrangements are diffeomorphic.

射影平面上的三角排列

Triangular arrangements on the projective plane

研究三角排列的组合学性质，证明其总能由单位根排列实现，并给出自由与非自由三角排列的示例。

On the structure of conjugation-free fundamental groups of conic-line arrangements

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkable properties: either it is a direct sum of free groups and a free abelian group, or it has a conjugation-free geometric presentation. In this paper, we first give a complete proof to the following key lemma: if we draw a new line through only one intersection point of a given re…

Conjugation-free geometric presentations of fundamental groups of arrangements

We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free geometric presentation: A real arrangement L, whose graph of multiple points is a union of disjoint cycles, has no line with more than two multiple points, and where the multiplicities of the multiple points are arbitrary. We also compute the exact group structure (by means of a semi-direct product of groups) of the arrangement of 6 lines whose graph consists of a …

实超平面排列的模2上同调环

Mod 2 magnitude cohomology ring of real hyperplane arrangements

本文针对有限中心实超平面排列的顶图，改进Koizumi的模长同调结果，利用典范上同调基确定其交叉分次模长上同调环，明确了直和项的秩性质。