Characterize perfect matchings and uniform suspension periods for bounded-jump circulant independence complexes

Determine which vertex orders yield perfect discrete Morse matchings for independence complexes of circulant graphs with a fixed finite bounded positive jump set, and determine when local duplication identities produce a uniform ordinary suspension period.

Background

For every fixed finite bounded positive jump set, the paper constructs a finite-state description and a rational generating series for critical faces of an ordered-toggle matching. This counts Morse cells but does not determine their attaching maps or the resulting homology and homotopy types.

For the jump set {2,3}, the paper proves an eight-periodic ordinary suspension pattern, while also showing that no uniform representation-suspension refinement exists for all common cyclic subgroups. The broader classification of perfect matching orders and uniform suspension mechanisms remains unresolved.