Seamed Riemann Surfaces

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 Consider a seamed orbifold Riemann surface constructed as follows. Start with the face poset of a three-dimensional cube. Let $\mathcal A_3$ denote the set of antipodal vertex pairs $$a=\{v,v^*\}.$$ To each such pair $a\in\mathcal A_3$, associate a spherical $2$-orbifold $$S^2(\alpha,\alpha)$$ whose two cone points are attached to the antipodal vertices $v$ and $v^*$. Equivalently, and more naturally from the complex-analytic point of view, we regard these spherical orbifolds as copies of $\mathbb{CP}^1$ and glue them along a common seam complex $\Gamma$. Thus the complex foam is $$\mathcal I_{\mathbb C}=\left(\bigsqcup_{a\in\mathcal A_3}^{4}\mathbb{CP}^1_a\right)\big/\!\sim_\Gamma .$$ The cubical scaffold should be understood primarily as an organizational device. Rather than constructing the foam by gluing disks one at a time, as in more traditional constructions of seamed Riemann surfaces or Klein foams, the cube records the incidence structure in advance: whenever two sheets o...
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A Real World Application of $\mathcal F$-completions

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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.


 





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