Short proofs of Tverberg-type theorems for cell complexes

We present short proofs of Tverberg-type theorems for cell complexes by S. Hasui, D. Kishimoto, M. Takeda, and M. Tsutaya. One of them states that for any prime power $r$, any complex $X$ topologically homeomorphic to $S^{(d+1)(r-1)-1}$, and any continuous map $f:X\to {\mathbb{R}}^d$ there are pairwise disjoint faces ${\sigma }_1,\dots ,{\sigma }_r$ of $X$ such that $f({\sigma }_1)\cap \dots f({\sigma }_r)\mathrm{\ne }\mathrm{\varnothing }$.