A Real World Application of $\mathcal F$-completions

The following diagram is a high explosive lens mold from the Manhattan project, drawn in secret by David Greenglass, to pass on to the Soviets:

Below is a 3D symmetrical extension of this high explosive lens mold:

This is precisely $\mathcal I$ which is a special subset of $CX_V$ in dimension $n=3$. You can see the outer colored loops that wrap around $\mathcal I$. This is a subset of $\Gamma$. This also gives a clean example of the coloring function $\phi: E \to \mathcal C$. 

The flat local system $\mathcal L$ lives on these colored loops and it is inherited directly from the geometry of $\mathcal I$.

Thus, we can form the enrichment $\mathcal X=(\phi,\mathcal L, \rho).$ For more details on this see the previous post: https://jzdynamics.blogspot.com/2025/04/the-geometry-of-delta-groupoid.html. 

Without going into too much detail on the physics - the right thing to do is to take geodesic flows along the white strands and add (i.e. direct sum) the force vectors restricted to the collision interfaces. This encodes the force vectors into a flat bundle or local system giving the net force bundle. We can now interpret $\rho$ (the holonomy representation) in terms of a flat connection (i.e. no curvature is detected). This is because any parallel transport along any colored loop path comprising a subset of $\Gamma$ yields no angle defect.

 

The Geometry of the Delta Groupoid

This post serves to marry the last three posts into one cohesive framework. The connection to foliational completions is also made clear.

Enriched Graphs and Delta Groupoid

Let $\Gamma = (V, E)$ be a finite connected tailless regular planar graph or multigraph, embedded in $\mathbb{R}^3$. Define an enrichment of $\Gamma$ to be a triple:

$$\mathcal{X} = (\phi, \mathcal{L}, \rho)$$

where:

- $\phi: E \to \mathcal{C}$ is a coloring or stratification into a finite set of colors $\mathcal{C}$,

- $\mathcal{L}$ is a local system (functor from the path groupoid of $\Gamma$ to $\mathrm{Vect}_k$), assumed flat,

- $\rho: \pi_1(\Gamma) \to \mathrm{Aut}(\mathcal{L})$ is a holonomy representation.

Let $\mathcal{E}_\Gamma$ denote the set of all such enrichments.

Define the Delta groupoid $\mathcal{D}_\Gamma$ as follows:

- Objects: enrichments $\mathcal{X} \in \mathcal{E}_\Gamma$.

- Morphisms: invertible maps

  $$\delta: (\phi, \mathcal{L}, \rho) \longrightarrow (\phi', \mathcal{L}', \rho')$$

  satisfying:

- The underlying graph $\Gamma$ is fixed:

$$Z_\Gamma(u) = Z_{\delta(\Gamma)}(u)$$

- The coloring transforms via pullback:

$$\phi' = \delta^* \phi$$

- The local system transforms functorially:

$$\mathcal{L}' = \delta^* \mathcal{L}, \quad \rho' = \rho \circ \delta_*$$

- Composition: $(\delta_2 \circ \delta_1)(\mathcal{X}) = \delta_2(\delta_1(\mathcal{X}))$.

- Each object has an identity, and morphisms are invertible.

Morphisms $\delta$ preserve $Z_{\Gamma}(u)$ but generally do not preserve $\phi$, $\mathcal{L}$, or $\rho$. Thus, $\delta$ encodes invariance of cycle structure and variance of enrichment data.

Given $\mathcal{X}, \mathcal{X}' \in \mathcal{E}_\Gamma$ with $\delta(\mathcal{X}) = \mathcal{X}'$, an important question is:

$$H^1(\Gamma, \mathcal{L}) \cong H^1(\Gamma, \mathcal{L}') \quad \text{?}$$

Suppose:

- $\delta$ is a geometric Delta morphism (e.g., rotation about $z=0$),

- $\Gamma \subset \mathbb{R}^3$ lies in $z=0$ plane,

- $\delta$ acts as an automorphism of $\Gamma$ and $\mathcal{L}$ preserving flatness.

Then:

$$\rho \simeq \rho \circ \delta_*, \quad H^1(\Gamma, \mathcal{L}) \cong H^1(\Gamma, \delta^*\mathcal{L}).$$

Such $\delta$ form a distinguished subgroupoid preserving the full enrichment.

Foliational Completions and the Interface Structure.

Let $X$ be a smooth, open, geodesically convex $n$-dimensional manifold with regular polytope boundary, and $V = \{v_i\}$ a finite set of vertices.

Define the block:

$$\mathcal{B} := X \cup \partial X$$

A foliational completion (or $\mathcal{F}$-completion) is given by foliations $\mathcal{F}_{v_{ij}}$ for each distinct pair $(v_i, v_j)$, satisfying:

$$\lim_{p \to \partial X} L_\alpha(p) = \{v_i, v_j\}$$

for leaves $L_\alpha$ of $\mathcal{F}_{v_{ij}}$.

The $\mathcal{F}$-completion is:

$$CX_V := \bigcup_{(v_i, v_j) \in V \times V} \mathcal{F}_{v_{ij}}$$

The Interface Structure $\mathcal{I}$

Inside $CX_V$ in dimension $n=3$, define $\mathcal{I}$ as the union of four symmetrical maximal surfaces of revolution inside the unit cube, each having constant Gaussian curvature.

Slicing $\mathcal{I}$ with planes $x_1, x_2, x_3 = 1/2$ yields two overlapping oval curves per slice, with intersections forming pieces of the graph $\Gamma$.

Thus, $\mathcal{I}$ geometrically generates $\Gamma$, tying together the foliational and graph-based frameworks.

Summary

The enriched Delta groupoid $\mathcal{D}_\Gamma$ and the foliational Interface $\mathcal{I}$ are intertwined:

- $\mathcal{I}$ provides a natural geometric source of $\Gamma$,

- $\mathcal{D}_\Gamma$ captures symmetries, preserving cycle structure but transforming local systems and cohomology.

Together, they describe a system where coarse spectral data remains invariant while finer enrichment data evolves under geometric transformations.

Delta Groupoid Associated to an Enriched Graph

 Let $\Gamma = (V, E)$ be a finite connected tailless regular planar graph or multigraph embedded in $\mathbf R^3$. Define an enrichment of $\Gamma$ to be a triple:

$$\mathcal{X} = (\phi, \mathcal{L}, \rho)$$

where, $\phi: E \to \mathcal{C}$ is a coloring or stratification of the edge set into a finite set of colors $\mathcal{C}$. $\mathcal{L}$ is a local system (i.e., a functor from the path groupoid of $\Gamma$ to $\mathrm{Vect}_k$). It is assumed a priori that connections are flat. $\rho: \pi_1(\Gamma) \to \mathrm{Aut}(\mathcal{L})$ is a holonomy representation.

Let $\mathcal{E}_\Gamma$ denote the set of all such enrichments. The Delta groupoid $\mathcal{D}_\Gamma$ is defined as follows:

Objects are enrichments $\mathcal{X} = (\phi, \mathcal{L}, \rho) \in \mathcal{E}_\Gamma$. Morphisms are invertible maps

$$\delta: (\phi, \mathcal{L}, \rho) \longrightarrow (\phi', \mathcal{L}', \rho')$$

that satisfy the following properties:

The underlying graph $\Gamma$ is fixed, thus the Ihara zeta function is preserved:

    $$Z_\Gamma(u) = Z_{\delta(\Gamma)}(u)$$

The coloring is transformed by a pullback or relabeling:

    $$\phi' = \delta^* \phi$$

The local system is transformed functorially:

    $$\mathcal{L}' = \delta^* \mathcal{L}, \quad \rho' = \rho \circ \delta_*$$

Composition is defined by:

$$(\delta_2 \circ \delta_1)(\mathcal{X}) = \delta_2(\delta_1(\mathcal{X}))$$

Each object has an identity morphism $\mathrm{id}_{\mathcal{X}}$. Every morphism $\delta \in \mathrm{Hom}(\mathcal{X}, \mathcal{Y})$ has an inverse $\delta^{-1} \in \mathrm{Hom}(\mathcal{Y}, \mathcal{X})$.

As can be seen above, the morphisms $\delta$ preserve $Z_{\Gamma}(u),$ but in general do not preserve $\mathcal L,\phi, \rho.$ In essence, $\delta$ "sees" both the invariance of $Z_{\Gamma}(u)$ and the variance of $\mathcal L,\phi,\rho$. This is because $Z_{\Gamma}$ is insensitive to $\delta$. $Z_{\Gamma}$ is a spectral object that detects cycles, not the higher enrichment data. Thus a given $\delta$ can transform an initialized bundle with flat connection into a bundle whose curvature is not zero.

Given two enrichments $\mathcal{X}, \mathcal{X}' \in \mathcal{E}_\Gamma$ such that $\delta(\mathcal{X}) = \mathcal{X}'$, under what conditions does

$$H^1(\Gamma, \mathcal{L}) \cong H^1(\Gamma, \mathcal{L}')$$

hold?

Let $\mathcal{X} = (\phi, \mathcal{L}, \rho) \in \mathcal{E}_\Gamma$, and suppose $\delta \in \mathcal{D}_\Gamma$ is a Delta morphism defined by a geometric transformation, such as a rotation about the normal to the $z = 0$ plane.

Assume further that:

- $\delta$ is a graph automorphism of $\Gamma \subset \mathbb{R}^3$, where $\Gamma$ is embedded in the $z = 0$ plane.

- $\delta$ also acts as an automorphism of the local system $\mathcal{L}$, in the sense that it maps fibers linearly and preserves the parallel transport structure.

- The connection on $\mathcal{L}$ is flat, and the orientation of vectors in $\mathcal{L}$ is compatible with the planar embedding.

Then $\delta$ preserves not only the Ihara zeta function $Z_\Gamma(u)$, but also the holonomy representation $\rho$ (up to isomorphism), and the twisted cohomology:

$$\rho \simeq \rho \circ \delta_*, \qquad H^1(\Gamma, \mathcal{L}) \cong H^1(\Gamma, \delta^* \mathcal{L}).$$

In particular, such a $\delta$ defines a morphism in a distinguished subgroupoid of $\mathcal{D}_\Gamma$ that preserves all enrichment data. 

However, in general, $\delta$ will only preserve $Z_{\Gamma},$ not $\phi,\mathcal L,\rho$.

Gamma sets and coloring

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.