7 problems

For an integer $n>4$, let $S^n$ denote the $n$-dimensional sphere, and say that it has a unique smooth structure when every smooth manifold homeomorphic to $S^n$ is diffeomorphic t…

Infinite-value conjecture. There is a suitable homeomorphism type on which Perelman's invariant takes on infinitely many distinct values.

Let $(X_{0},X_{1})$ be an exotic pair of closed, simply-connected smooth $4$-manifolds, meaning that they are homeomorphic but not diffeomorphic. Let $X\#Y$ denote the connected su…

Trisection genus invariance conjecture. If two smooth $4$-manifolds $X$ and $Y$ are exotic, then

Let $C=C(r,s;m)$ be Gompf's cork, and define the twisted double … Here $r,s,m$ are integers with $r,s>0>m$, and $k$ is a nonzero integer. Gompf's conjecture. The homotopy 4-sphere…

Let $M\to B$ and $M'\to B$ be tangentially homeomorphic smooth manifold bundles with common fiber a closed oriented even-dimensional manifold, and let $\Theta(M',M)$ denote their s…

Infinite exotic smooth structures conjecture. There are infinitely many mutually nondiffeomorphic $4$-manifolds homeomorphic to $X$.