Political Structures and the Topology of Simplicial Complexes

The paper’s Theorem 5.18 states that if some Betti number β_n is nonzero, then P contains β_n subsets of agents (each of size at least n+2) that are not viable, and in the weighted model these subsets are not w-viable for any w; hence P itself is not w-viable for any w. The proof given selects β_n independent n-cycles and notes their vertex sets cannot be simplices because a simplex is acyclic; the weighted conclusion follows from the monotonic (downward-closed) nature of w-viability (the definitions appear in the weighted model subsection) . The paper works with reduced homology (so β_0 means components minus one), which resolves n=0 edge cases . The model’s solution gives a different, more quantitative proof via the long exact sequence of the pair (Δ, P) and a counting argument on missing faces, yielding t ≥ β_n (and for n=0, t ≥ β̃_0), where t is the number of non-viable (n+2)-subsets; this immediately implies the same qualitative conclusions and the weighted corollary. Thus both are correct, with the model providing a sharper bound not present in the paper’s sketch. The paper’s own proof sketch appears in the text adjoining Theorem 5.18 and aligns with these conclusions .