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Acyclic component hypotheses for total cut complexes of disconnected graphs

Manuscript

Main result

Let $G=G_1\sqcup\cdots\sqcup G_k$ be a finite simple graph with $k$ nonempty connected components and $n$ vertices. For $d\geq2$, let $\Delta_d^t(G)$ be the total $d$-cut complex. If $k\geq d$ and, for every component $G_i$ and every $2\leq r\leq d$, the complex $\Delta_r^t(G_i)$ is void or integer-acyclic, then

$$ \Delta_d^t(G)\simeq \bigvee_{\binom{k-1}{d-1}} S^{n-d-1}. $$

The theorem covers the complete component range $k\geq d$. It subsumes the previous v0.2-beta result for the three low-component layers and answers Question 30 of Carnero Bravo for the component class stated above by removing the component clique-complex simple-connectivity condition.

Proof structure

  • An acyclic composition diagram and its homology-colimit spectral sequence compute the integral homology in every component range.
  • For $k\geq d+1$, direct combinatorial connectivity arguments upgrade the homology calculation to the asserted homotopy type.
  • For the boundary layer $k=d$, a weak-composition cover has contractible nonbaseline blocks and connected intersections; successive van Kampen arguments close the fundamental group.
  • The graph $C_4\sqcup K_1$ at $d=2$ supplies a negative control showing that component acyclicity cannot simply be omitted.

Scope and prior-work boundary

This version is a successor to The Low-Component Cases for Total Cut Complexes of Disconnected Graphs. It uses the composition-poset and homotopy-colimit framework of Carnero Bravo, while replacing objectwise contractibility by integral acyclicity and supplying new connectivity arguments under the weaker hypotheses.

The paper does not classify arbitrary disconnected graphs, determine the simple-homotopy type, or remove every component acyclicity hypothesis. The weakening from contractible to integer-acyclic is formal until a graph-domain strictness witness is known. Absolute priority remains qualified pending broader public and expert review.

This is a theoretical proof. Finite computation was used only for exploratory falsification and is not part of the proof.

AI-assisted research disclosure

Carptopus is the responsible author. OpenAI Codex was used as an AI-assisted research, verification, and writing tool. The manuscript contains the full disclosure and responsibility statement.

Keywords

Total cut complex; bounded-independence complex; disconnected graph; simplicial complex; Alexander duality; homotopy colimit; van Kampen theorem; wedge of spheres; graph complex; algebraic topology.