Spin Lifts and Relational Stability of Subgroup Intersections

This paper studies a simple but important discontinuity phenomenon in the space of closed subgroups of a compact Lie group. In SO (3), the paper considers a fixed subgroup, the rotation group SO (2) around the z-axis, and a one-parameter family of order-two subgroups generated by 180-degree rotations around slightly tilted axes. Although these subgroups vary continuously in the Chabauty topology, their intersections with the fixed SO (2) subgroup do not vary continuously. At the special aligned position, the intersection is isomorphic to Z/2, but after any nonzero tilt, the intersection collapses to the trivial group. Thus, an arbitrarily small change in the subgroup can produce a sudden loss of relational intersection structure. The main observation is that this collapse becomes more stable after lifting the configuration through the double cover SU (2) → SO (3). In SO (3), the intersection drops from Z/2 to the trivial group. In SU (2), however, the lifted intersection drops only from Z/4 to Z/2. The surviving Z/2 is exactly the spin kernel ±I. Therefore, the spin cover does not remove the Chabauty discontinuity, but it prevents the intersection from disappearing completely. The kernel of the covering map acts as a persistent residual intersection. The paper formulates this as a general persistence principle for finite covering homomorphisms. For any covering map, the intersection of the lifted subgroups equals the lift of the original intersection. When the intersections are finite, the size of the lifted intersection is multiplied by the size of the kernel. Consequently, even if the original intersection becomes trivial in the base group, the lifted intersection still contains at least the kernel of the cover. This gives a precise sense in which finite covers preserve a minimal layer of relational structure. The paper also compares this phenomenon with infinitesimal rigidity. It directly computes that the first Lie algebra cohomology of the maximal torus SO (2) inside SO (3), with coefficients in the quotient representation, vanishes. This means that the subgroup itself is infinitesimally rigid: every first-order deformation is absorbed by conjugation. However, this rigidity of a single subgroup does not imply stability of intersections between subgroups. The example shows that individual rigidity and relational Chabauty stability are different notions. In summary, the paper shows that subgroup relations can be more fragile than the subgroups themselves. A subgroup may deform smoothly, and even be infinitesimally rigid, while its intersection with another subgroup changes discontinuously. Passing to the spin cover reveals a persistent kernel that survives this collapse. The spin kernel can therefore be interpreted as a minimal residual structure that remains when relational intersection data degenerates.