Topology of independence complexes and cycle structure of hypergraphs

TL;DR

The paper extends the Kalai–Meshulam framework from graphs to hypergraphs by proving that hypergraphs with no Berge cycles of length divisible by $3$ have independence complexes $I(H)$ with total reduced Betti numbers summing to at most $1$, highlighting a deep link between forbidden cycle structures and topological simplicity. Central to the approach is a hypergraph star cluster theorem: for a suitable vertex $v$, $I(H)$ is homotopy equivalent to a suspension of a localized intersection of stars around $v$ and the incident edges, enabling recursive reductions. The work introduces an explicit star-dissolution construction, structural conditions for the theorem, and a family of Betti-number bounds when the hypergraph contains only a bounded number of disjoint ternary Berge cycles, using edge-star expansions and Erdős–Pósa-type results for modulo-3 cycles. These results connect topological properties of independence complexes to combinatorial cycle constraints and provide a unified framework that extends known graph results to hypergraphs. The findings have implications for understanding when independence complexes are contractible or spheres and for translating hypergraph cycle structure into quantitative topological invariants.

Abstract

Recently, Zhang and Wu proved a conjecture of Kalai and Meshulam, showing that for every graph $G$ without induced cycles of length divisible by $3$, the sum of all reduced Betti numbers of its independence complex $I(G)$ is at most $1$. We extend this result to the hypergraph setting. Namely, we show that the same conclusion holds for any hypergraph $H$ that does not contain a Berge cycle of length divisible by $3$. This establishes a broader connection between forbidden cycle structures and the topological simplicity of independence complexes. As a key tool, we introduce a hypergraph analogue of Barmak's star cluster theorem for graphs. This new theorem implies, in particular, that if a hypergraph $H$ has a vertex $v$ that is not isolated and is not contained in an induced Berge cycle of length $3$, then there exists a hypergraph $H'$ with fewer vertices than $H$ such that the independence complex of $H$ is homotopy equivalent to the suspension of the independence complex of $H'$.

Figures (5)

• Figure 1: By applying Theorem \ref{['thm:structure']} to $H$ and $v$, we obtain that $H_v$ is isomorphic to $K_{2,2}$.
• Figure 2: We obtain $H_v$ by applying Theorem \ref{['thm:structure']} for a $3$-uniform hypergraph $H$ and $v$.
• Figure 3: For a $3$-uniform tight path $P_{6,3}$, we obtain $\mathcal{L}_3(P_{6,3})=P_{10,3}$.
• Figure 4: The edge-star expansion $H_e$ at $e=\{v_1,v_2,v_3,v_4\}$.
• Figure 5: Although $H$ has no ternary Berge cycle, $H_v$ contains a triangle.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2: Star cluster theorem
• Theorem 1.3
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• proof
• Theorem 3.1: Barmak, Star cluster theorem for graphs
• Theorem \ref{thm:main}
• Lemma 3.2