An acyclic $d$-partition of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$

In this paper we introduce a $d$-partition ${\mathcal{E}}_d^{(r)}=({\mathrm{\Omega }}_1^{(r,d)},{\mathrm{\Omega }}_2^{(r,d)},\dots ,{\mathrm{\Omega }}_d^{(r,d)})$ of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$. We prove that ${\mathcal{E}}_d^{(r)}$ is homogeneous and that each hypergraph ${\mathrm{\Omega }}_i^{(r,d)}$ is acyclic (i.e. has zero Betti numbers). As an application, we show that the map $de{t}^{S^r}$ is nontrivial for every $r$, which gives a partial answer to a conjecture from [14].