Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homology inclusion of complex line arrangements

Adrien Rodau

TL;DR

This work develops a refined invariant for complex line arrangements by studying the homology inclusion from the boundary manifold into the exterior, leveraging a graph-manifold (Waldhausen) decomposition and a combinatorial graph stabiliser that captures generator differences across graphed embeddings. By encoding ordered graphed embeddings with a graph ordering $W$ and quotienting by difference maps, the invariant $\mathcal{J}_W(A)$ becomes independent of embedding choices while remaining sensitive to topological embeddings, enabling distinctions between Zariski pairs that share the same combinatorics. The framework unifies and extends prior invariants like the loop-linking number, providing a finite presentation for the graph stabiliser and a practical computation route via wiring diagrams or braid monodromy, implemented in Sage. Applications include new oriented Zariski quadruplets (and their unoriented counterparts), including explicit examples with 11- and 13-line configurations, illustrating finer discrimination power than existing invariants. The results advance the understanding of how boundary-exterior interactions encode embedding information, offering potential extensions to twisted homology and broader classes of complex plane curves.

Abstract

We introduce a new topological invariant of complex line arrangements in $\mathbb{CP}^2$, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski pairs which have the same combinatorics but different embeddings. Building on ideas developed by B. Guerville-Ballé and W. Cadiegan-Schlieper, we consider the inclusion map of the boundary manifold to the exterior and its effect on homology classes. A careful study of the graph Waldhausen structure of the boundary manifold allows to identify specific generators of the homology. Their potential images are encoded in a group, the graph stabiliser, with a nice combinatorial presentation. The invariant related to the inclusion map is an element of this group. Using a computer implementation in Sage, we compute the invariant for some examples and exhibit new Zariski pairs.

Homology inclusion of complex line arrangements

TL;DR

This work develops a refined invariant for complex line arrangements by studying the homology inclusion from the boundary manifold into the exterior, leveraging a graph-manifold (Waldhausen) decomposition and a combinatorial graph stabiliser that captures generator differences across graphed embeddings. By encoding ordered graphed embeddings with a graph ordering and quotienting by difference maps, the invariant becomes independent of embedding choices while remaining sensitive to topological embeddings, enabling distinctions between Zariski pairs that share the same combinatorics. The framework unifies and extends prior invariants like the loop-linking number, providing a finite presentation for the graph stabiliser and a practical computation route via wiring diagrams or braid monodromy, implemented in Sage. Applications include new oriented Zariski quadruplets (and their unoriented counterparts), including explicit examples with 11- and 13-line configurations, illustrating finer discrimination power than existing invariants. The results advance the understanding of how boundary-exterior interactions encode embedding information, offering potential extensions to twisted homology and broader classes of complex plane curves.

Abstract

We introduce a new topological invariant of complex line arrangements in , derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski pairs which have the same combinatorics but different embeddings. Building on ideas developed by B. Guerville-Ballé and W. Cadiegan-Schlieper, we consider the inclusion map of the boundary manifold to the exterior and its effect on homology classes. A careful study of the graph Waldhausen structure of the boundary manifold allows to identify specific generators of the homology. Their potential images are encoded in a group, the graph stabiliser, with a nice combinatorial presentation. The invariant related to the inclusion map is an element of this group. Using a computer implementation in Sage, we compute the invariant for some examples and exhibit new Zariski pairs.
Paper Structure (22 sections, 41 theorems, 114 equations, 11 figures)

This paper contains 22 sections, 41 theorems, 114 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Combinatorics of complex line arrangements
  3. Generalities
  4. Orderings
  5. Boundary manifold
  6. Circle bundles
  7. Graph structure
  8. Flat stars
  9. Linear and circular stars
  10. Mapping class groups
  11. Model disc
  12. Action on flat stars
  13. Graphed embeddings
  14. Homology inclusion
  15. Presentations of the homology groups
  16. ...and 7 more sections

Key Result

lemma 1

The right action of $\mathfrak{S}_m$ on $\mathfrak{L}\left(W\right)$ respects the left quotient $c : \mathfrak{L}\left(W\right) "->" \mathfrak{C}\left(W\right)$.

Figures (11)

  • Figure 1: Incidence graph of the MacLane combinatorics
  • Figure 2: Star orderings
  • Figure 3: Examples of flat stars
  • Figure 4: Standard linear star $\bar{'a}_0$ and curve diagram $'g_0$ on $'D_m$
  • Figure 5: Stripe $N_{\bar{'a}}$ of a star $\bar{'a}$
  • ...and 6 more figures

Theorems & Definitions (99)

  • definition 1
  • definition 2
  • definition 3
  • remark 2.1
  • lemma 1
  • proof
  • definition 4
  • lemma 2
  • proof
  • theorem 3.1
  • ...and 89 more