A Local Block Criterion for Periodic Morse Reductions of Circulant Independence Complexes

Abstract

For G_n = Cay(Z/nZ, {±2, ±3}), we determine the homotopy type of its independence complex for every n ≥ 6. If n = 8q + r, the answer is a wedge of five (2q − 1)-spheres for r = 0, one (2q − 1)-sphere for r = 1, 2, 3, three 2q-spheres for r = 4, and one (2q + 1)-sphere for r = 5, 6, 7. Consequently, Ind(G_{n+8}) is homotopy equivalent to the double suspension of Ind(G_n). The proof introduces a local block-duplication criterion that turns a finite boundary-state identity for any bounded jump set into an infinite periodic discrete-Morse reduction. Its {2, 3} instance proves the complete classification. We also determine the rational character of cyclic translation on the unique nonzero reduced homology. For a prime companion benchmark, we identify the integral top homology with the augmentation ideal of Z[C_13] and describe its rationally acyclic cyclic quotient. Exact certificate programs and regression tests are included in the source archive. This deposit contains the 13-page preprint and its 17-file source and certificate archive.