5 problems

Benson–Carlson conjecture. Any finite group of rank $r$ acts freely on a finite CW-complex homotopy equivalent to

For each positive integer $n$, define … Let $X$ be a finite CW-complex whose mod-$2$ cohomology is that of $\prod_{i=1}^k\mathbb{R}P^{n_i}$. Suppose that $G=(\mathbb{Z}/2)^r$ acts…

Let $\Delta$ be an orientable $\mathbbm{k}$-homology manifold of dimension $d-1$ admitting a free action by the cyclic group $\mathbb{Z}/p\mathbb{Z}$. For the finer…

Homotopy free rank conjecture. For groups whose rank is greater than one, $\operatorname{rk} G=\operatorname{hfrk} G$.

Free homologically trivial action conjecture. Every finite group acts freely and homologically trivially on a product of spheres.