Kirby Problem 4.112

Definitions and mathematical statement

A trisection of $S^4$ is a decomposition into three smooth four-dimensional one-handlebodies $X_i\cong\natural^{k_i}(S^1\times B^3)$, whose pairwise intersections are genus-$g$ three-dimensional handlebodies and whose triple intersection is a closed genus-$g$ surface. The standard genus-zero trisection has three four-balls, pairwise three-balls, and central sphere.

An elementary $i$-stabilization takes connected sum of a trisection with the standard genus-one trisection of $S^4$ of type $(1;1,0,0)$, with the entry $1$ in sector $i$. It increases $g$ and $k_i$ by one and leaves the other two $k_j$ unchanged. Geometrically it transfers a neighborhood of a boundary-parallel arc to that sector. Permuting sector names gives the three elementary kinds.

Is every positive-genus trisection of the standard smooth four-sphere equivalent to one obtained from the genus-zero trisection by finitely many such elementary stabilizations? Equivalence means a diffeomorphism of $S^4$ carrying the sector decomposition to the other, with a consistent allowance for permutation of sector labels.

All trisections are included, balanced or unbalanced. If using only balanced stabilizations, group one stabilization of each kind together and keep track of the resulting restriction. The problem concerns destabilizing an arbitrary given trisection all the way to genus zero; the existence of some common stabilization with a standard trisection does not establish this direction.

Short English statement

Is every trisection of the standard four-sphere obtained by stabilizing its genus-zero trisection?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2988 (KP-4.112), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.112. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.