9 problems

Let $\Delta^4$ be a Poincare ball, and let $\Delta^5$ be a contractible 5-manifold. A handle decomposition has only 0-, 1- and 2-handles when all its handles have these indices. 0-…

A Gluck ball is the Poincare 4-ball associated with a Gluck-twisted $S^4$. Gluck ball 5-ball conjecture. If $\Delta^4$ is a Gluck ball, then … This is described as a special case o…

A Poincare 4-ball is a contractible 4-manifold whose boundary is $S^3$. A Schoenflies 4-ball is a Poincare 4-ball that embeds in $S^4$. Poincare ball 5-ball conjecture. If…

Let $\Delta$ be a Schoenflies ball, and let $-\Delta$ denote its complementary Schoenflies ball in $S^4$. Write $\bar\Delta$ for $\Delta$ with the opposite orientation. Negative-eq…

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…

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

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

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