Morse Theoretic Templates for High Dimensional Homology Computation

The paper proves that Mate(·, α) is acyclic on any H_n-embeddable complex via an α-stability/most-preferred-bit zig-zag argument (Theorem 1.2 and its proof, together with the pullback via incidence embeddings and the Mate recursion) . It then assembles fiberwise matchings into a global acyclic matching using the Patchwork Theorem (Theorems 1.3 and 2.12) . The candidate solution proves the same claim by identifying Mate with a lexicographic first-available-coordinate matching on each fiber and invoking Babson–Hersh for acyclicity there, before applying Patchwork. This yields the same conclusion. The only minor imprecision is the statement that w_{≤i} “coincides with α_i on A_{i−1}”; strictly, by the Mate definition (Eqn. (4)), it coincides with α_i only on those elements of A_{i−1} whose mates also lie in A_{i−1} . Otherwise, both arguments are correct and consistent with the paper’s framework.