Gamma sets and coloring

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Let $\Gamma = (V, E)$ be a finite, planar labeled, undirected, tailless, topological multigraph ($4$-regular) arising from a $z=0$ slice of a stratified foliated manifold. The graph admits a decomposition into two overlapping subgraphs ("ovals") $\mathcal{O}_1, \mathcal{O}_2 \subseteq \Gamma$, such that their union reconstructs $\Gamma$:

$$\Gamma = \mathcal{O}_1 \cup \mathcal{O}_2, \quad \text{with } \mathcal{O}_1 \cap \mathcal{O}_2 \neq \varnothing.$$

We define a coloring function

$$\phi: E \to \mathcal{C} = \{c_1, c_2\}$$

that assigns each edge a domain label corresponding to one of the two ovals. This coloring encodes the stratified foliation structure by partitioning edge interactions into two local holonomy domains.


Each edge $e \in E$ carries a local holonomy map $\tau_e \in \mathrm{Aut}(\mathcal{F})$, where $\mathcal{F}$ is a coefficient system (e.g., a local system or vector bundle). For any based cycle $\gamma = (v_0 \to e_1 \to \cdots \to e_k \to v_0)$, the holonomy representation is given by the composition

$$\rho_\phi(\gamma) = \tau_{e_k} \circ \cdots \circ \tau_{e_1},$$

where the twists $\tau_{e_i}$ are interpreted according to the domain color $\phi(e_i)$.


This coloring data induces a twisted cohomology group:

$$H^1_\phi(\Gamma, \mathcal{F}),$$

which encodes cocycle compatibility conditions modulated by the domain decomposition $\phi$.


Let $r$ be a reflection operation across a plane of symmetry (e.g., slicing through two opposing vertices). This acts geometrically on $\Gamma$ and induces a transformation of the coloring function:

$$\phi_r := \phi \circ r.$$

The reflection $r$ not only reverses the geometric orientation of one oval (say, $\mathcal{O}_1$), but also induces a permutation of color domains:

$$\phi_r(e) \neq \phi(e), \quad \text{in general}.$$

In fact, multiple reflections may yield nontrivial compositions of color swaps, resulting in a sequence of colorings:

$$\phi, \quad \phi_r, \quad \phi_{r \circ r'}, \quad \ldots$$

where each composition alters the assignment of domains in increasingly nontrivial ways.


Let $\mathcal{G}$ denote the group generated by these geometric reflection-induced color permutations acting on $\phi$. Then:


- The graph $\Gamma$ remains fixed (up to label-preserving isomorphism),

- The coloring function $\phi$ varies by $\phi \mapsto g \cdot \phi$, for $g \in \mathcal{G}$,

- The holonomy representation changes: $\rho_{\phi} \neq \rho_{g \cdot \phi}$,

- The twisted cohomology groups are inequivalent:

    $$H^1_\phi(\Gamma, \mathcal{F}) \not\cong H^1_{g \cdot \phi}(\Gamma, \mathcal{F}) \quad \text{in general}.$$


The action of $\mathcal{G}$ partitions the space of colorings into symmetry classes. Each orbit of $\phi$ under $\mathcal{G}$ defines a distinct equivalence class of stratified foliated structures sharing the same underlying graph and (Ihara) zeta function, but with dynamically inequivalent holonomy and cohomology.

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