Acyclic Component Hypotheses for Total Cut Complexes of Disconnected Graphs

Abstract

Let G be a finite simple graph with k nonempty connected components and n vertices, and let d be at least 2. We prove that if k is at least d and every relevant component total-cut complex is void or integer-acyclic, then the total d-cut complex of G is homotopy equivalent to a wedge of binomial(k-1,d-1) spheres of dimension n-d-1. The theorem covers the complete range k at least d, subsumes the preceding low-component result, and answers Question 30 of Carnero Bravo for this component class by removing the component clique-complex simple-connectivity condition. The proof combines an acyclic composition diagram, a homology-colimit spectral sequence, direct connectivity arguments, and a weak-composition cover with van Kampen at the boundary k=d. A C4 disjoint-union K1 example shows that component acyclicity cannot simply be omitted. Status: Public Beta; internally verified candidate proof; external mathematical review pending.