The homology groups of finite cyclic covering of line arrangement complement
Yongqiang Liu, Wentao Xie
TL;DR
This work extends the study of the first homology of finite cyclic coverings of line arrangement complements by proving torsion-freeness under Cohen–Dimca–Orlik–type non-vanishing conditions and deriving explicit divisibility bounds for the Alexander polynomial. By constructing and analyzing a boundary manifold $X$ around a fixed line, the authors compute the fundamental group and Fox-calculus-based Alexander matrix to relate $H_1(\mathcal{U}^{\varphi,N},\mathbb{K})$ to $H_1(X^{\varphi,N},\mathbb{K})$, yielding a dimension formula $\dim H_1(\mathcal{U}^{\varphi,N},\mathbb{K}) = n-1+\sum (m_i-2)(\gcd(\varepsilon_i,N)-1)$. Under the assumption $\epsilon=1$ and $\varepsilon_i\neq 0$ for all $m_i>2$, and with $\gcd(\varepsilon_i,N)=1$, the paper shows $H_1(\mathcal{U}^{\varphi,N},\mathbb{Z}) \cong \mathbb{Z}^{n-1}$, extending Williams' bound from Milnor fibers to general complex line arrangements. The results provide divisibility relations for twisted and untwisted Alexander polynomials and offer structural insight into when finite covers have torsion-free first homology, with Milnor fiber arising as a key special case.
Abstract
In this paper, we study the first homology group of finite cyclic covering of complex line arrangement complement. We show that this first integral homology group is torsion-free under certain condition similar to the one used by Cohen-Dimca-Orlik. In particular, this includes the case of the Milnor fiber, which generalizes the previous results obtained by Williams for complexified line arrangement to any complex line arrangement.