12 problems

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

Let $S$ be a complex algebraic surface homeomorphic to … The preceding result states that all algebraic surfaces homeomorphic to $3\mathbb{C}P^{2}\#18\overline{\mathbb{C}P^{2}}$ ar…

The constructions described produce families of homeomorphic simply connected $4$-manifolds with identical Seiberg–Witten invariants. Nondiffeomorphism conjecture. No two of these…

Fintushel–Stern injectivity conjecture. The association

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…

A genus function is an invariant associated with a smooth 4-manifold that records the minimal genera of embedded surfaces representing its homology classes. Consider simply connect…

Let $X$ be an oriented smooth $4$-manifold, let $k$ be an integer, and let $\eta\in \mathcal{A}^{\operatorname{even}}_{k}$ be nonzero. Denote by $E^{\eta}_{X}\to S^k$ the smooth bu…

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

A pair of smooth, closed 4-manifolds $X_1$ and $X_2$ is exotic if the two manifolds are homeomorphic but not diffeomorphic. In the oriented case, a single stabilization means takin…

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 $X$.