9 problems

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 $\pi$ be a finite group that acts freely on $S^{2n-1}$ for some $n$. An action on an odd-dimensional sphere is almost linear if its restriction to every cyclic or $2$-hyperelem…

Spherical space-form conjecture. For other quotients of the sphere $\Gamma\backslash S^n$,

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 $\Sigma$ be a spherical space form, and let $\operatorname{Isom}(\Sigma)$ and $\operatorname{Diff}(\Sigma)$ denote its isometry and diffeomorphism groups. Hatcher's conjecture.…

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 $d$ be an odd integer with $d\geq 5$. An isospectral and non-isometric pair of $d$-dimensional spherical space forms is said to have highest volume if its common volume is maxi…

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…

Let $S^n$ be the round $n$-sphere and let $\Gamma$ be a finite group acting freely and isometrically on $S^n$, so that $S^n/\Gamma$ is a finite quotient of the round sphere. Let…