Homology of Local Systems on Real Line Arrangement Complements
Abstract
We study the homology groups of the complement of a complexified real line arrangement with coefficients in complex rank-one local systems. Using Borel--Moore homology, we establish an algorithm computing their dimensions via the real figures of the arrangement. It enables us to give a new upper bound. We further consider the case where the arrangement contains a sharp pair and make partial progress on a conjecture proposed by Yoshinaga.
Summary
This paper investigates the homology groups of the complement of a complexified real line arrangement, focusing on computations with complex rank-one local systems. The main problem addressed is whether the Betti numbers of the complement, with coefficients in a complex rank-one local system, are determined by the arrangement's intersection lattice and monodromy map. The authors introduce an algorithm to compute the dimensions of these homology groups using Borel-Moore homology and the real figures of the arrangement. This approach is presented as a dual version of Yoshinaga's method. A key result is a new upper bound on the first Betti number, h1(M(A), L), expressed in terms of the number of resonant points on a line. They also consider the specific case where the arrangement contains a sharp pair of lines, making progress towards a conjecture by Yoshinaga. The paper's findings contribute to the ongoing effort to understand how combinatorial data governs the topological invariants of hyperplane arrangement complements, an area with connections to monodromy on cohomology of Milnor fibrations. The paper builds upon existing work by Orlik and Solomon, Cohen and Suciu, Yoshinaga, and others. It provides a novel algorithm based on Borel-Moore homology, which contrasts with previous approaches using Fox calculus or twisted minimal chain complexes. The authors explicitly describe generators of the first homology group in terms of angles at resonant points and relations derived from bounded chambers in the real arrangement. The paper also offers a detailed analysis of the coefficients involved in these relations and provides an explicit computation for a reflection arrangement of type A3. The results provide an improved upper bound on h1(M(A), L) compared to previous work, especially for arrangements containing a sharp pair.
Key Insights
- •The paper presents a new algorithm for computing the first homology group of the complement of a complexified real line arrangement using Borel-Moore homology. This algorithm can be seen as a dual version of Yoshinaga’s approach.
- •The authors provide a combinatorial upper bound on h1(M(A), L) for complexified real line arrangements: h1(M(A), L) ≤ max(0, #R0 - 1), where R0 is the set of resonant points on a line l0.
- •For arrangements containing a "sharp pair," the authors prove that h1(M(A), L) ≤ 1, generalizing a result by Yoshinaga.
- •In the case of arrangements with a sharp pair and a constant monodromy map ζ (where ζ is a primitive d-th root of unity and 2 divides d), the authors show that H1(M(A), Lζ) = 0, making partial progress on a conjecture.
- •The paper provides a detailed description of the generators of H1(M(A), L) in terms of "angles" at resonant points (Definition 3.1) and relations derived from bounded chambers (Definition 3.3).
- •Lemma 3.8 provides an explicit formula for computing the coefficients λp(∆) which are crucial for determining relations in the homology group.
- •The paper includes an explicit example (Example 3.10) for a reflection arrangement of type A3, demonstrating the algorithm and the sharpness of the upper bound.
Practical Implications
- •The algorithm developed in this paper can be implemented to compute the homology groups of real line arrangement complements, aiding in the study of their topological properties.
- •The theoretical results and the algorithm can be used by researchers in algebraic topology, singularity theory, and hyperplane arrangement theory.
- •The explicit upper bounds on the Betti numbers can serve as a benchmark for evaluating the complexity of these computations and for designing more efficient algorithms.
- •The paper opens up avenues for future research, including further progress on Yoshinaga's conjecture and extending the results to more general hyperplane arrangements.
- •The techniques used in this paper, such as the application of Borel-Moore homology and the analysis of resonant points, can be adapted to study other topological invariants of hyperplane arrangements.