27 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…

Meier's conjecture. If $X$ admits a genus-three trisection, then $X$ is diffeomorphic to a spun lens space $S_p$ or its sibling $S_p'$, $S^4$, or a connected sum of copies of…

Let $(X,\omega)$ be a symplectic $4$--manifold. A Weinstein trisection of $(X,\omega)$ is the notion proposed as an appropriate generalization of the standard trisection of…

Trisection conjecture for $S^4$. Every trisection of $S^4$ is isotopic to a stabilization of the genus 0 trisection.

A smooth 4-manifold is said to admit a handle decomposition without 1- or 3-handles if it has a handle decomposition containing no handles of index 1 or 3. Handle decomposition con…

Kashaev's invariant conjecture. The invariants satisfy

Kashaev-invariant relation. The trisection invariant associated to the Hopf triplet in the cited construction should be related to Kashaev's invariant.

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 $18$.

A trisection is a decomposition of a smooth 4-manifold into three 4-dimensional 1-handlebodies satisfying the standard pairwise and triple-intersection conditions. Let $S^4$ denote…

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

Let $\mathcal{T}$ be a trisection of a closed, oriented $4$-manifold. Write $L_{\mathcal{T}}$ for its Kirby–Thompson length, and let $L$ denote the minimum of $L_{\mathcal{T}}$ ove…

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…

No 1-handles conjecture. Every simply-connected 4-manifold has a handle decomposition with no 1-handles.

Trisected 11/8-conjecture. One has

Stein trisection conjecture. Every projective complex surface admits a Stein trisection.

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…

For a lens space $L(p,q)$, let $n_1(p,q)$ be the smallest integer such that $L(p,q)$ embeds almost in the spine of a minimal-genus trisection of $\#^{n_1} S^2 \times S^2$, and let…

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.

Let $X^4$ be a smooth $4$-manifold equipped with a trisection, and let $\Sigma$ and $\Sigma'$ be isotopic embedded surfaces in $X^4$. Consider bridge trisections of $\Sigma$ and…