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The critical group of a combinatorial map

Criel Merino, Iain Moffatt, Steven Noble

TL;DR

This work extends the classical sandpile (critical) group from abstract graphs to graphs embedded in orientable surfaces, defining a map-level critical group $K(\mathbb{G})$ via the cokernel of a topological analogue of the cycle–cocycle matrix, $A(\vec{\mathbb{G}},T)+I$. It establishes multiple equivalent formulations, including a Laplacian-based definition on the directed medial graph and a chip-firing interpretation on map edges, and proves foundational properties such as well-definedness, plane-map compatibility with the classical group, and a Matrix--Quasi-tree Theorem that counts spanning quasi-trees. The theory shows that $K(\mathbb{G})$ mutates as the embedding changes, acting as a perturbation of the underlying graph’s classical critical group by topological data, while retaining a robust combinatorial core tied to spanning quasi-trees and delta-matroid insights. Concrete computations on several canonical embedded graphs illustrate the differences from the classical case and highlight practical methods for determining the group structure.

Abstract

Motivated by the appearance of embeddings in the theory of chip firing and the critical group of a graph, we introduce a version of the critical group (or sandpile group) for combinatorial maps, that is, for graphs embedded in orientable surfaces. We provide several definitions of our critical group, by approaching it through analogues of the cycle-cocycle matrix, the Laplacian matrix, and as the group of critical states of a chip firing game (or sandpile model) on the edges of a map. Our group can be regarded as a perturbation of the classical critical group of its underlying graph by topological information, and it agrees with the classical critical group in the plane case. Its cardinality is equal to the number of spanning quasi-trees in a connected map, just as the cardinality of the classical critical group is equal to the number of spanning trees of a connected graph. Our approach exploits the properties of principally unimodular matrices and the methods of delta-matroid theory.

The critical group of a combinatorial map

TL;DR

This work extends the classical sandpile (critical) group from abstract graphs to graphs embedded in orientable surfaces, defining a map-level critical group via the cokernel of a topological analogue of the cycle–cocycle matrix, . It establishes multiple equivalent formulations, including a Laplacian-based definition on the directed medial graph and a chip-firing interpretation on map edges, and proves foundational properties such as well-definedness, plane-map compatibility with the classical group, and a Matrix--Quasi-tree Theorem that counts spanning quasi-trees. The theory shows that mutates as the embedding changes, acting as a perturbation of the underlying graph’s classical critical group by topological data, while retaining a robust combinatorial core tied to spanning quasi-trees and delta-matroid insights. Concrete computations on several canonical embedded graphs illustrate the differences from the classical case and highlight practical methods for determining the group structure.

Abstract

Motivated by the appearance of embeddings in the theory of chip firing and the critical group of a graph, we introduce a version of the critical group (or sandpile group) for combinatorial maps, that is, for graphs embedded in orientable surfaces. We provide several definitions of our critical group, by approaching it through analogues of the cycle-cocycle matrix, the Laplacian matrix, and as the group of critical states of a chip firing game (or sandpile model) on the edges of a map. Our group can be regarded as a perturbation of the classical critical group of its underlying graph by topological information, and it agrees with the classical critical group in the plane case. Its cardinality is equal to the number of spanning quasi-trees in a connected map, just as the cardinality of the classical critical group is equal to the number of spanning trees of a connected graph. Our approach exploits the properties of principally unimodular matrices and the methods of delta-matroid theory.
Paper Structure (12 sections, 36 theorems, 73 equations, 4 figures)

This paper contains 12 sections, 36 theorems, 73 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The classical critical group
  4. Combinatorial maps
  5. Matrices
  6. The definition of the critical group of a map, and a summary of results
  7. The group is well-defined
  8. Connections with the classical critical group
  9. Enumerating quasi-trees and the order of the critical group
  10. The critical group through the Laplacian
  11. Chip firing
  12. Further examples

Key Result

Theorem 2.1

Let $G$ be a connected, loopless abstract graph with vertex $q$. Then the number of spanning trees of $G$ is equal to $\det(\Delta^q(G))$.

Figures (4)

  • Figure 1: Moving between maps and embedded graphs.
  • Figure 2: The partial dual of an edge in a directed rotation system.
  • Figure 3: Maps for Examples \ref{['basicexamp1']}, \ref{['basicexamp2']}, \ref{['ex:lap']} and \ref{['ex:cf']}.
  • Figure 4: Maps, drawn here as embedded graphs, considered in Section \ref{['sec:examples']}. (The orientation is anticlockwise.)

Theorems & Definitions (69)

  • Theorem 2.1: Matrix--Tree Theorem
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.5
  • proof
  • Example 2.6
  • Definition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Theorem 3.4
  • ...and 59 more