5 problems

Montgomery–Yang problem. Every pseudofree $\mathbb{S}^1$-action on $\mathbb{S}^5$ has at most $3$ non-free orbits.

Boundary-determination conjecture. If $\partial X$ and $\partial X'$ are diffeomorphic, then $X$ and $X'$ are diffeomorphic.

Let $L$ be a 5-dimensional rational homology sphere with $H_1(L,\mathbb Z)=0$ admitting a fixed-point-free differentiable circle action. For an orbit $O$, define its exceptional mu…

Let $L$ be a 5-dimensional rational homology sphere with $H_1(L,\mathbb Z)=0$ admitting a pseudo-free differentiable circle action. Let $O_1,\dots,O_k$ be the nonfree orbits with s…

Sasaki–Einstein existence conjecture. For each integer $k\leq 8$ and $n\geq 2$, the Smale $5$-manifold