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Partially multiplicative quandles and simplicial Hurwitz spaces

Andrea Bianchi

TL;DR

This work develops partially multiplicative quandles (PMQs), unifying partial monoids and quandles and establishing an algebraic framework with completions, enveloping groups, and PMQ-group pairs. It then constructs simplicial Hurwitz spaces $Hur^{\Delta}(\cal Q)$ by passing to crossed objects in braided monoidal categories, connects classical Hurwitz spaces via the PMQ $\mathcal{Q}=G\sqcup\{\mathbf{1}\}$, and realises a relative cellular chain complex through a double bar construction. The second major thread treats simplicial Hurwitz spaces, proving that the associated cellular complexes coincide with reduced total complexes of bisimplicial modules, and that Poincaré and Koszul properties arise in favorable PMQs. The paper culminates with a detailed study of geodesic PMQs from symmetric groups, computing enveloping groups, describing completions, and establishing foundational Koszulity results that motivate future links to moduli spaces and topological field-theoretic viewpoints. Together, these results set up a robust algebraic-topological apparatus for generalized Hurwitz spaces and their homological invariants, with anticipated applications to moduli spaces of Riemann surfaces and beyond.

Abstract

We introduce partially multiplicative quandles (PMQ), a generalisation of both partial monoids and quandles. We set up the basic theory of PMQs, focusing on the properties of free PMQs and complete PMQs. For a PMQ $\mathcal{Q}$ with completion $\hat{\mathcal{Q}}$, we introduce the category of $\hat{\mathcal{Q}}$-crossed topological spaces, and define the Hurwitz space $\mathrm{Hur}^Δ(\mathcal{Q})$: it is a $\hat{\mathcal{Q}}$-crossed space, and it parametrises $\mathcal{Q}$-branched coverings of the plane. The definition recovers classical Hurwitz spaces when $\mathcal{Q}$ is a discrete group $G$. Finally, we analyse the class of PMQs $\mathfrak{S}_d^{\mathrm{geo}}$ arising from the symmetric groups $\mathfrak{S}_d$, and we compute their enveloping groups and their PMQ completions.

Partially multiplicative quandles and simplicial Hurwitz spaces

TL;DR

This work develops partially multiplicative quandles (PMQs), unifying partial monoids and quandles and establishing an algebraic framework with completions, enveloping groups, and PMQ-group pairs. It then constructs simplicial Hurwitz spaces by passing to crossed objects in braided monoidal categories, connects classical Hurwitz spaces via the PMQ , and realises a relative cellular chain complex through a double bar construction. The second major thread treats simplicial Hurwitz spaces, proving that the associated cellular complexes coincide with reduced total complexes of bisimplicial modules, and that Poincaré and Koszul properties arise in favorable PMQs. The paper culminates with a detailed study of geodesic PMQs from symmetric groups, computing enveloping groups, describing completions, and establishing foundational Koszulity results that motivate future links to moduli spaces and topological field-theoretic viewpoints. Together, these results set up a robust algebraic-topological apparatus for generalized Hurwitz spaces and their homological invariants, with anticipated applications to moduli spaces of Riemann surfaces and beyond.

Abstract

We introduce partially multiplicative quandles (PMQ), a generalisation of both partial monoids and quandles. We set up the basic theory of PMQs, focusing on the properties of free PMQs and complete PMQs. For a PMQ with completion , we introduce the category of -crossed topological spaces, and define the Hurwitz space : it is a -crossed space, and it parametrises -branched coverings of the plane. The definition recovers classical Hurwitz spaces when is a discrete group . Finally, we analyse the class of PMQs arising from the symmetric groups , and we compute their enveloping groups and their PMQ completions.

Paper Structure

This paper contains 48 sections, 42 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Outline of the article
  4. Motivation
  5. A brief history of Hurwitz spaces
  6. A brief history of configuration spaces with summable labels
  7. Acknowledgments
  8. Algebraic theory of partially multiplicative quandles
  9. Partially multiplicative quandles and their relation to groups
  10. Basic definitions and first examples
  11. Enveloping group of a PMQ
  12. Adjoint action and PMQ-group pairs
  13. Complete PMQs
  14. Free groups and associated PMQs
  15. Free sub-PMQs
  16. ...and 33 more sections

Key Result

Proposition 1

The canonical map $\mathcal{Q}\to\hat{\mathcal{Q}}$ from a PMQ to its completion is injective.

Figures (1)

  • Figure 1: A configuration in $\mathrm{Hur}^{\Delta}(\mathcal{Q})$ of the form $(\underline{a};\underline{s},\underline{t})$, where $\underline{a}\in\mathrm{Arr}_{2,3}(\mathcal{Q})$ is an admissible array of size $4\times 5$ whose only entries in $\mathcal{Q}_+$ are $a_{1,1}$, $a_{1,2}$, $a_{2,2}$ and $a_{2,3}$. We have $I(\underline{a})=\left\{(1,1),(1,2),(2,2),(2,3)\right\}$, and $P=\left\{z_{1,1},z_{1,2},z_{2,2},z_{2,3}\right\}$. The monodromy $\psi$ associates with the element $[\gamma_{i,j}]\in\pi_1(\mathbb{C}\setminus P;*)$ the element $a_{i,j}\in\mathcal{Q}_+$. Since $[\gamma]=[\gamma_{2,2}]^{([\gamma_{1,1}][\gamma_{1,2}])^{-1}}$ in $\pi_1(\mathbb{C}\setminus P;*)$, the monodromy $\psi$ associates with $[\gamma]$ the element $a_{2,2}^{([a_{1,1}][a_{1,2}])^{-1}}\in\mathcal{Q}$, where we use the adjoint action of $\mathcal{G}(\mathcal{Q})$ on $\mathcal{Q}$.

Theorems & Definitions (145)

  • Proposition 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Lemma 5
  • Proposition 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 135 more