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Monodromy of supersolvable toric arrangements

Christin Bibby, Daniel C. Cohen, Emanuele Delucchi

TL;DR

The paper develops a topological framework for supersolvable abelian and toric arrangements by embedding their complements into towers of fiber bundles pulled back from classical configuration-space bundles. In the toric setting, the monodromy of supersolvable bundles factors through the Artin braid group, while strictly supersolvable cases factor further through the pure braid group, enabling explicit computations of fundamental groups, lower central series Lie algebras, and cohomology rings. A key tool is Hansen's polynomial viewpoint, which yields coefficient and root maps that realize the bundles as pullbacks of configuration-space bundles, and in turn provide direct descriptions of invariants such as Koszulity of cohomology and the topological complexity. The work offers detailed treatments of rank-two circuits and type C toric arrangements, illustrating the general theory and enabling explicit, computable presentations for the invariants of the complements. Overall, the results connect combinatorial supersolvability to concrete topological and algebraic structures via configuration-space monodromy and braid-group representations, with implications for linearity and computational topology of these spaces.

Abstract

We study topological aspects of supersolvable abelian arrangements, toric arrangements in particular. The complement of such an arrangement sits atop a tower of fiber bundles, and we investigate the relationship between these bundles and bundles involving classical configuration spaces. In the toric case, we show that the monodromy of a supersolvable arrangement bundle factors through the Artin braid group, and that of a strictly supersolvable arrangement bundle factors further through the Artin pure braid group. The latter factorization is particularly informative -- we use it to determine a number of invariants of the complement of a strictly supersolvable arrangement, including the cohomology ring and the lower central series Lie algebra of the fundamental group.

Monodromy of supersolvable toric arrangements

TL;DR

The paper develops a topological framework for supersolvable abelian and toric arrangements by embedding their complements into towers of fiber bundles pulled back from classical configuration-space bundles. In the toric setting, the monodromy of supersolvable bundles factors through the Artin braid group, while strictly supersolvable cases factor further through the pure braid group, enabling explicit computations of fundamental groups, lower central series Lie algebras, and cohomology rings. A key tool is Hansen's polynomial viewpoint, which yields coefficient and root maps that realize the bundles as pullbacks of configuration-space bundles, and in turn provide direct descriptions of invariants such as Koszulity of cohomology and the topological complexity. The work offers detailed treatments of rank-two circuits and type C toric arrangements, illustrating the general theory and enabling explicit, computable presentations for the invariants of the complements. Overall, the results connect combinatorial supersolvability to concrete topological and algebraic structures via configuration-space monodromy and braid-group representations, with implications for linearity and computational topology of these spaces.

Abstract

We study topological aspects of supersolvable abelian arrangements, toric arrangements in particular. The complement of such an arrangement sits atop a tower of fiber bundles, and we investigate the relationship between these bundles and bundles involving classical configuration spaces. In the toric case, we show that the monodromy of a supersolvable arrangement bundle factors through the Artin braid group, and that of a strictly supersolvable arrangement bundle factors further through the Artin pure braid group. The latter factorization is particularly informative -- we use it to determine a number of invariants of the complement of a strictly supersolvable arrangement, including the cohomology ring and the lower central series Lie algebra of the fundamental group.

Paper Structure

This paper contains 28 sections, 17 theorems, 91 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Overview and structure of the paper
  4. Acknowledgements
  5. Arrangements and configuration spaces
  6. Abelian and toric arrangements
  7. Supersolvability
  8. Somewhat ordered configuration spaces
  9. Artin representation
  10. Abelian arrangement bundles as pullbacks
  11. Toric arrangements
  12. Toric arrangement bundles
  13. Polynomials
  14. Fundamental Group
  15. Strictly supersolvable toric arrangements
  16. ...and 13 more sections

Key Result

Lemma 2.2.3

Let $\mathscr{A}$ be an abelian arrangement in $T\cong\mathbb{G}^d$, and suppose that $\mathscr{P}(\mathscr{A}_Y)$ is a corank-one M-ideal of $\mathscr{P}(\mathscr{A})$. Let $p: T\to T/Y$ be the projection to the quotient. Then for every $X\in \mathscr{P}(\mathscr{A})$ we have $p(X)\in \mathscr{P}(\

Figures (5)

  • Figure 1: See \ref{['ex:A', 'ex:A2', 'ex:A3']}, and \ref{['subsec:rank2circ']}.
  • Figure 2: Pullback diagram of \ref{['thm:pullback']}
  • Figure 3: Pullback diagram of \ref{['thm:toricpullback']}
  • Figure 4: Loops in $M(\mathscr{B})$ when $n=6$, $m=3$.
  • Figure 5: Loops and paths in $M(\mathscr{C}_1)$

Theorems & Definitions (62)

  • Definition 2.1.1
  • Remark 2.1.2
  • Remark 2.1.3
  • Example 2.1.4
  • Definition 2.2.1
  • Remark 2.2.2
  • Lemma 2.2.3
  • proof
  • Example 2.2.4
  • Proposition 2.3.1
  • ...and 52 more