17 problems

A trisection of a closed 4-manifold is a decomposition into three 4-dimensional 1-handlebodies whose pairwise intersections are 3-dimensional handlebodies and whose triple intersec…

Kashaev's invariant conjecture. The invariants satisfy

Let $a4n$ act smoothly and homologically trivially on $a4cp^2$, and let an equivariant trisection be a trisection preserved by the action. Homologically trivial linearization conje…

Let $a4n$ act smoothly on $S^4$, and let $S$ be its fixed-point set, assumed to be an unknotted $2$-sphere. An equivariant bridge trisection is a bridge trisection preserved by the…

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

Connected-sum pants-invariant conjecture. The $\mathcal{L}^*_3$-invariant of $Y_m$ is equal to $9m$.

Spun-lens-space conjecture. The $\mathcal{L}^*_3$-invariant of spun lens spaces is equal to $18$.

Waldhausen-type conjecture. Every trisection of $S^4$ is isotopic to either the genus $0$ trisection or its stabilization.

Let $S$ be a properly embedded smooth surface in $B^4$. Genus-zero bridge trisection conjecture. Every properly embedded smooth surface $S$ in $B^4$ has a genus $0$ bridge trisecti…

A closed smooth $4$-manifold is said to admit a genus-three trisection when it admits a trisection of genus three. A $4$-manifold is a spun lens space when it is obtained by spinni…

Trisected 11/8-conjecture. One has

Efficient trisection conjecture. Every simply-connected, complex four-manifold admits an efficient trisection.

Let $M$ be an orientable $3$-manifold. Universal embedding conjecture. There is a positive integer $n$ such that $M$ embeds almost in the spine of a minimal-genus trisection of … T…

Let $M$ be a closed orientable $3$-manifold that is not of the form $\#^k S^1 \times S^2$, and let $M$ embed so as to almost lie in the spine of a genus-$2$ trisection of a $4$-man…

Meier's conjecture. Every trisection of $S^4$ is either the $(0,0)$-trisection or a stabilization of the $(0,0)$-trisection.

Trisection-genus-three classification conjecture. Every irreducible 4-manifold with trisection genus three is either the spin of a lens space, or a Gluck twist on a specific 2-knot…

Let $\mathcal{T}$ be a trisection of the standard smooth four-sphere $\mathcal{S}^4$. Let $\mathcal{S}^0$ denote its standard $(0,0)$-trisection, and let…