5 problems

Let $G$ be a finite group, and let $G$ act freely on … The rank $r(G)$ is the maximum of the $p$-ranks $r_p(G)$ over primes $p$ dividing $|G|$, where $r_p(G)$ is the largest $r$ su…

Let $Q_{8m}$ denote the generalized quaternion group of order $8m$ acting freely on $S^3$, and let $r$-th sequential topological complexity be denoted by $\mathsf{TC}_r$. Sequentia…

Let $G$ be a finite group. Define its rank by … and define $h(G)$ to be the least integer $k$ such that $G$ acts freely on a product $S^{n_1}\times\cdots\times S^{n_k}$ of spheres.…

Let $p$ be a prime, let $n_1,\ldots,n_k$ be positive integers, and let $(\mathbb{Z}_p)^r$ act freely on the product $S^{n_1}\times\cdots\times S^{n_k}$. Free-rank conjecture for pr…

Let $M^3$ be a closed three-manifold with positive Ricci curvature, and let $\Sigma$ be an orientable minimal surface in $M^3$. Schoen's conjecture. If $\Sigma$ has index one, then…