GLMY Fundamental Group of Graphs

• The GLMY fundamental group is an algebraic invariant from path homology that encodes combinatorial path equivalence and cycle structures in graphs.
• It admits equivalent descriptions via gain graph presentations and the closed neighborhood complex, aligning with classical cycle spaces and first path homology groups.
• Its sensitivity to graph curvature leads to finiteness results under positive Bakry–Émery conditions, bridging discrete geometry with topological invariants.

The Grigor'yan–Lin–Muranov–Yau (GLMY) fundamental group of a graph is an algebraic invariant arising from the path homology theory of finite and infinite graphs. It generalizes the classical notion of the fundamental group from algebraic topology to the discrete setting of graphs, capturing higher-order connectivity and cycle structure by encoding combinatorial path equivalence in terms of local moves. This fundamental group plays a pivotal role in the interaction between homotopy, path homology, gain graphs, graph coverings, and the cycle spaces of graphs, and admits several equivalent descriptions, notably as the fundamental group of the closed neighborhood complex and via canonical gain group presentations. The GLMY fundamental group is tightly related to the algebraic structure of the first path homology group and is sensitive to graph curvature, with profound finiteness implications under positive Bakry–Émery curvature.

1. Formal Definition and Path Homology Presentation

Let $G=(V,E)$ be a (simple) undirected graph, with base vertex $v_0$. For homotopy purposes, $G$ is viewed as having arcs $x \to y$ and $y \to x$ for each edge $\{x, y\}\in E$. A based loop of length $n$ is a sequence $(v_0,v_{i_1},\ldots,v_{i_n}=v_0)$ such that consecutive vertices are adjacent or identical.

Two based loops are declared GLMY-homotopic (C-homotopic) if they differ by a finite sequence of the following elementary moves (and their inverses):

• (a) Remove a pause: $(\ldots,a,a,\ldots) \mapsto (\ldots,a,\ldots)$
• (b) Delete a backtrack: $(\ldots,a,b,a,\ldots) \mapsto (\ldots,a,\ldots)$ whenever $a \sim b$
• (c) Triangle contraction: $(\ldots,a,b,c,\ldots) \mapsto (\ldots,a,c,\ldots)$ when $\{a,b,c\}$ is a 3-cycle
• (d) Square contraction: $$(\ldots,a,b,c,d,\ldots) \mapsto (\ldots,a,d,\ldots)$$ when $(a \to b \to c \to d \to a)$ is a 4-cycle

The set of equivalence classes of based loops under these moves forms the GLMY fundamental group $\pi_1^{\mathrm{GLMY}}(G,v_0)$ under the operation of concatenation. This construction coincides with the "closed 2-fundamental group" defined via path homology in other works (Kempton et al., 2017).

2. Gain Graph and Edge Presentation

An alternative description of $\pi_1^{\mathrm{GLMY}}(G,v_0)$ is given by a gain-graph presentation. Given a spanning tree $T\subset G$ and a collection $B$ of circuits (for GLMY, $B$ is the set of all 3- and 4-cycles), define the group

$$\Gamma(G,T;B) = \left\langle e_{xy} \mid e_{yx}=e_{xy}^{-1},\, e_{uv}=1 \, (uv\in T),\, \prod_{i=1}^{\ell} e_{u_i u_{i+1}} = 1 \, (\text{for each } (u_1,\ldots,u_\ell)\in B) \right\rangle$$

where $e_{xy}$ represents the generator corresponding to the directed edge $x\to y$. The correspondence between paths and edge words, respecting the defining relations, gives a canonical isomorphism

$\pi_1^{\mathrm{GLMY}}(G,v_0) \cong \Gamma(G,T;B)$

In this model, cycle relations in $B$ and tree trivializations reflect the combinatorial path homotopy moves (Kempton et al., 2017).

3. Relationship to the Closed Neighborhood Complex

The GLMY fundamental group is naturally isomorphic to the fundamental group of the closed neighborhood complex $\mathcal{N}[G]$ of $G$. Here, $\mathcal{N}[G]$ is the simplicial complex whose simplices are finite vertex sets contained in the closed neighborhood $N_G[v] = \{w \mid (v, w) \in E\} \cup \{v\}$ for some $v\in V(G)$. The main result [Theorem B in (Matsushita, 16 Nov 2025)] is

$$\pi_1(\mathcal{N}[G], v) \cong \pi_1^{\mathrm{GLMY}}(G, v)$$

The proof constructs a bijection by thickening each simplicial edge to a two-step path through an appropriate vertex in $G$, ensuring the allowable homotopies in the edge-path group correspond precisely to GLMY's spike and square moves. Thus, the path-homological fundamental group of a graph gains a purely simplicial-topological realization.

4. Abelianization and Connection to Path Homology

The abelianization of $\pi_1^{\mathrm{GLMY}}(G, v_0)$ is canonically isomorphic to the first path homology group $H_1^{\mathrm{path}}(G; \mathbb{Z})$, as established by Kempton–Münch–Yau (Kempton et al., 2017). Formally,

$$\pi_1^{\mathrm{GLMY}}(G, v_0)^{\mathrm{ab}} \cong H_1^{\mathrm{path}}(G; \mathbb{Z})$$

This isomorphism sends the class of a loop to the corresponding chain in $\operatorname{Ker}\partial_1$. The GLMY abelianization quotients the integer cycle space $Z_1(G; \mathbb{Z})$ by the submodule generated by 3- and 4-cycles, reflecting the fact that triangles and squares are contractible in this setting. This correspondence makes the GLMY group a combinatorial refinement of the classical cycle space and path homology, with explicit algebraic realization.

5. Impact of Graph Curvature: Finiteness Results

Fundamental geometric results link the GLMY group to Bakry–Émery curvature conditions. For a finite graph $G$ satisfying the curvature-dimension condition $\mathrm{CD}(K, \infty)$ with $K > 0$, there is no infinite cover of $G$ that preserves all 3- and 4-cycles (Kempton et al., 2017). Consequently, $\pi_1^{\mathrm{GLMY}}(G, v_0)$ is a finite group. The analogy is direct with the classical Bonnet–Myers theorem for manifolds and is established by showing that covers inherit curvature and are diameter-bounded by Liu–Yau's estimate. This affirms that positive curvature eliminates nontrivial infinite "looping" phenomena in the sense of the GLMY fundamental group.

6. Illustrative Examples and Applications

• For the path graph $P_m$, $\mathcal{N}[P_m]$ is contractible, yielding $\pi_1^{\mathrm{GLMY}}(P_m) = 1$.
• For the cycle graph $C_n$, both the edge-path group of $\mathcal{N}[C_n]$ and $\pi_1^{\mathrm{GLMY}}(C_n)$ are isomorphic to $\mathbb{Z}$, matching the fundamental group of the circle.
• For the complete graph $K_m$, $\mathcal{N}[K_m] = K_m$ is simply connected for $m \ge 3$, hence $\pi_1^{\mathrm{GLMY}}(K_m) = 1$.

Additionally, Matsushita establishes a homotopy equivalence between the suspension of the closed neighborhood complex of the complement graph and the independence complex of the canonical double covering, and a combinatorial Alexander duality involving the neighborhood hypergraph (Matsushita, 16 Nov 2025). While not directly used in the proof of the group isomorphism, these connections embed $\mathcal{N}[G]$ and hence the GLMY group within a larger web of topological and combinatorial constructions.

7. Interaction with Digraph Homotopy and Exact Sequences

Extensions of the GLMY theory to digraphs admit further categorical structure, as in the construction of homotopy groups $T_n(G)$ for digraphs (Li et al., 2024). The fundamental group $T_1(G)$ is defined via C-homotopy classes of based loops, reflecting the GLMY presentation but sensitive to edge directions. There exists a long exact sequence (a digraph version of the Puppe sequence), generalizing classical results from topological spaces. The GLMY fundamental group here distinguishes directed phenomena invisible to the standard CW-complex realization. For instance, for a directed $n$-cycle, $T_1(C_n)\cong \mathbb{Z}$, as in the undirected case, but alternating oriented cycles may remain nontrivial under GLMY C-homotopy.

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Collectively, the Grigor’yan–Lin–Muranov–Yau fundamental group of a graph encodes nonabelian topological information adapted to the discrete graph context, unifying combinatorial, algebraic, and geometric concepts and revealing new curvature-sensitive structural results unattainable with classical invariants (Matsushita, 16 Nov 2025, Kempton et al., 2017, Li et al., 2024).