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 $σ_1,\ldots,σ_r$ of $X$ such that $f(σ_1)\cap\ldots f(σ_r)\ne\emptyset$.

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