论文标题
双重覆盖物和排名一$ \ mathbb {z} $的积分同源组 - 最小CW复合物的本地系统
Integral homology groups of double coverings and rank one $\mathbb{Z}$-local system for minimal CW complex
论文作者
论文摘要
令$ x $为连接的有限CW综合体。 $ x $的连接双重覆盖物由非零的共同体类别$ω\在h^1（x，x，\ mathbb {z} _2）$中分类。用$ x^ω$表示双重覆盖空间。存在相应的非平凡等级一$ \ mathbb {z} $ - 本地系统$ \ MATHCAL {l}_Ω$上的$ x $。 $ x^ω$的积分同源组与本地系统$ \ MATHCAL {l}_Ω$之间的共同同源组之间的关系是什么？当$ x $相当于最小CW综合体时，我们对此问题给出了完整的答案。特别是，这解决了Ishibashi，Sugawara和Yoshinaga最近提出的猜想，以补充超平面布置。作为一个应用程序，当$ x $是超平面布置补充时，$ \ mathcal {l}_Ω$满足某些条件时，我们表明$ h _*（x^ω，\ mathbb {z}）$是组合确定的。
Let $X$ be a connected finite CW complex. A connected double covering of $X$ is classified by a non-zero cohomology class $ω\in H^1(X,\mathbb{Z}_2)$. Denote the double covering space by $X^ω$. There exists a corresponding non-trivial rank one $\mathbb{Z}$-local system $\mathcal{L}_ω$ on $X$. What is the relation between the integral homology groups of $X^ω$ and the homology groups of the local system $\mathcal{L}_ω$? When $X$ is homotopy equivalent to a minimal CW complex, we give a complete answer to this question. In particular, this settles a conjecture recently proposed by Ishibashi, Sugawara and Yoshinaga for hyperplane arrangement complement. As an application, when $X$ is a hyperplane arrangement complement and $\mathcal{L}_ω$ satisfies certain conditions, we show that $H_*(X^ω,\mathbb{Z})$ is combinatorially determined.
