Topology of independence complexes and cycle structure of hypergraphs

TL;DR

This paper extends a topological bound on independence complexes from graphs to hypergraphs, linking forbidden cycle structures to topological simplicity, and introduces a hypergraph analogue of a key theorem.

Contribution

It generalizes a known graph result to hypergraphs by establishing a connection between cycle restrictions and the topology of independence complexes, and introduces a new hypergraph star cluster theorem.

Findings

The sum of Betti numbers of independence complexes is at most 1 for hypergraphs without Berge cycles of length divisible by 3.

Introduces a hypergraph analogue of Barmak's star cluster theorem.

Shows homotopy equivalence to suspensions under certain cycle restrictions.

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 $H'$.