The rational homology of independence complexes of Kneser graphs on triples

The independence complex of the Kneser graph $\text{KG}(n,k)$ is the simplicial complex of pairwise intersecting families of $n$-subsets of $[2n+k]$; equivalently, it is a Vietoris–Rips complex of the $n$-subsets of $[2n+k]$ under the symmetric-difference metric. Feng and Wang asked whether, for $n=3$, its reduced homology is nontrivial only in dimensions $6$ and $9$, and conjectured the rank in dimension $9$. We answer the question affirmatively over the rationals and determine all the associated Betti numbers. For every $k\ge 0$, the only nonzero reduced Betti numbers using rationals of $\text{Ind}(\text{KG}(3,k))$ are $${\beta }_6=\left(\genfrac{}{}{0px}{}{k+5}{7}\right)+29\left(\genfrac{}{}{0px}{}{k+6}{7}\right)\text{and}{\beta }_9=\left(\genfrac{}{}{0px}{}{k+6}{6}\right)\mathrm{.}$$ The proof uses an inductive decomposition based on a simplicial retraction and Barmak’s star clusters, Dowker duality together with Jonsson’s computation of the homology of complexes of graphs with bounded covering number, and a relative Mayer–Vietoris spectral sequence which reduces the problem to the six-element sets.