20 problems

Morse-Novikov additivity conjecture. One should have

Let $M$ be a $3$-manifold whose fundamental group admits a free product decomposition. A connected sum decomposition of $M$ induces a free product decomposition of its fundamental…

Gompf's irreducibility conjecture. In any smooth connected sum decomposition of $X$, one of the summands $X_1$ or $X_2$ is a homotopy sphere. This says that minimal symplectic four…

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…

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 $\mathcal{K}$ be a three-dimensional simplicial sphere, and let $\mathcal{Z}_{\mathcal{K}}$ be its moment-angle manifold. Suppose that either $\mathcal{K}^1$ is a chordal graph…

Lambert-Cole–Meier's conjecture. The trisection genus is additive under connected sum:

Let $K_1$ be a nontrivial prime knot with resultant unknot probability $\alpha$, and let $K_2$ be a nontrivial prime knot formed by recursive connected sums with $K_1$. Write…

Poincaré-conjecture consequence. The monoid ${\mathcal M^{\mathrm{Top}}_{n}}$ never has non-trivial units, and ${\mathcal M^{\mathrm{PL}}_{n}}$ has no non-trivial units except pote…

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…

Let $k_1$ and $k_2$ be prime knots, and let $t(k)$ denote the tunnel number of a knot $k$. Their connected sum is denoted by $k_1\# k_2$. A knot is meridionally primitive if it has…

Let $k_1$ and $k_2$ be knots, and let $t(k)$ denote the tunnel number of a knot $k$. Their connected sum is denoted by $k_1\# k_2$. A knot is meridionally primitive if it has the p…

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

Let $P$ be a simple polytope and let $Z^{\mathbf C}$ denote its associated complex moment-angle manifold. A polytope is dual-neighborly when every collection of at most $[d/2]$ fac…

Let $N_k$ be closed connected orientable $4$-manifolds, let $f_k,f_k':N_k\to\mathbb R^7$ be embeddings, and suppose $BH(f_k)=BH(f_k')=u_k$. Let $\eta_{u_k}(f_k,f_k')$ be the relati…