11 problems

Let $M_1$ and $M_2$ be closed, orientable $3$-manifolds, and let $H_i$ be a Heegaard surface in $M_i$ for $i=1,2$. The connected sum $H_1\# H_2$ is a Heegaard surface in…

Let $X$ be a smooth, compact, oriented, simply connected $4$-manifold without boundary. A connected sum here means a decomposition into smooth four-manifolds, and both orientations…

A smooth, compact, oriented $4$-manifold without boundary is understood as a connected sum of symplectic manifolds, allowing either the symplectic or the opposite orientation, and…

A smooth, closed and oriented $4$-manifold is a smooth manifold of dimension four with the stated properties. A symplectic manifold is a smooth manifold equipped with a symplectic…

A smooth, closed, oriented and simply connected $4$-manifold is a smooth manifold of dimension four with the stated properties. A symplectic manifold is a smooth manifold equipped…

Morse-Novikov additivity conjecture. One should have

Anti-conjecture to the Unknotting Additivity Conjecture. For every non-trivial knot $K$, there is a symbiont knot $K^\prime$ for which

Let $K\#K'$ be a connected sum of knots, let a collection of unknotting crossing arcs for $K\#K'$ be given, and let the 2-sphere specifying the connected sum be fixed. Crossing-arc…

Let $K_1$ and $K_2$ be ribbon knots, and let $r(K)$ denote the minimum ribbon number of a ribbon disk for $K$. Ribbon number additivity conjecture. If $K_1$ and $K_2$ are ribbon kn…

Let $T(2,3)$ denote the trefoil, let $(\sigma_1\sigma_2\sigma_3)^4\sigma_1\sigma_2$ denote the braid whose closure is used in the connected sum, and let $\#_m$ denote the $m$-fold…

Connected-sum monopole-class conjecture. The rational part of a monopole class of $N_1\#\cdots\#N_n$ is expressed as