5 problems

Product-structure conjecture. The product structure $\partial A_0\times I$ extends over $A$ if and only if $D$ is concordant in $B_{1/2}\times I$ to the boundary-connected sum of…

Let $p:M\to B$ and $p':M'\to B$ be smooth manifold bundles with closed even-dimensional fibers, with $M'$ an exotic smoothing of $M$, meaning that $M'$ is fiberwise tangentially ho…

Let $p:M\to B$ be a smooth manifold bundle with closed even-dimensional fibers, and consider stable exotic smoothings of such bundles after rationalization. Goette–Igusa's Rigidity…

Let $M$ be a closed, connected, smooth manifold of dimension at least $5$ with torsion-free fundamental group. An $h$-cobordism over $M$ is a smooth cobordism over $M$ for which bo…

Hatcher's conjecture. The map induces a rational isomorphism on $\pi_{4k}$ when $N$ is large compared to $k$.