Extension to higher-dimensional cellular sheaves

Extend the arithmetic persistence framework from graphs to cellular sheaves on higher-dimensional complexes by analyzing Bockstein spectral sequence behavior beyond E2, identifying the contributions of higher differentials, and characterizing interactions with cup products in determining torsion and barcode structure.

Background

The current results exploit the collapse of the Bockstein spectral sequence for 1-dimensional bases, making torsion explicit at the first page. In higher dimensions, nontrivial differentials and algebraic operations such as cup products may influence torsion and lifting obstructions.

A generalization would broaden applicability to higher-dimensional sheaf-theoretic settings, requiring careful homological analysis to recover barcode-like invariants of torsion across precision.

References

Several mathematical questions remain open. Extension to cellular sheaves of dimension greater than one involves additional subtleties from higher differentials in the Bockstein spectral sequence and potential interactions with cup products.

— Precision-Graded Cohomology and Arithmetic Persistence for Network Sheaves (2511.00677 - Ghrist et al., 1 Nov 2025) in Section 7 (Conclusion)