Is trisection genus additive under connected sum?

Every smooth closed 4-manifold can be cut into three simple pieces, a trisection, much as every closed 3-manifold has a Heegaard splitting. Heegaard genus adds under connected sum; does the analogous trisection genus? A positive answer would have striking consequences, including the smooth 4-dimensional Poincaré conjecture.

Problem

Gay and Kirby showed that every smooth, closed, oriented, connected 4-manifold $X$ has a trisection: a decomposition $X = X_1 \cup X_2 \cup X_3$ into three copies of a boundary connected sum of copies of $S^1 \times B^3$, meeting pairwise in 3-dimensional handlebodies, with triple intersection a closed surface. The trisection genus $g(X)$ is the smallest genus of that central surface over all trisections of $X$.

Question. Is trisection genus additive under connected sum, that is,

$$g(X \# Y) = g(X) + g(Y)$$

for all such 4-manifolds $X$ and $Y$?

The inequality $g(X \# Y) \le g(X) + g(Y)$ is easy, by gluing trisections; the question is whether genus can drop.

Background

The question appears in the problem list of the AIM workshop on trisections and low-dimensional topology, and was conjectured in print by Lambert-Cole and Meier. In dimension three, Heegaard genus is additive by Haken's lemma. The source notes the consequence for the Poincaré conjecture: if $X$ were a fake 4-sphere, then $X \# k(S^2 \times S^2)$ is diffeomorphic to $k(S^2 \times S^2)$ for large $k$, so additivity would force $g(X) = 0$, and the only 4-manifold of trisection genus zero is $S^4$. More generally, additivity would imply that homeomorphic smooth 4-manifolds have the same trisection genus (Lambert-Cole and Meier).

What is known

• Trisections exist and are unique up to stabilisation (Gay and Kirby, 2016).
• Meier and Zupan (2017) showed that the only closed 4-manifolds with a trisection of genus two are $S^2 \times S^2$ and the connected sums of two summands, each of which is $S^1 \times S^3$, $\mathbb{CP}^2$ or $\overline{\mathbb{CP}^2}$, each with a unique genus-two trisection up to diffeomorphism.
• Lambert-Cole and Meier (2022) conjectured additivity and showed that it implies equal trisection genus for exotic pairs of closed 4-manifolds.
• Dunfield, Kegel, Li and Ren (2026, preprint) proved that trisection genus is not computable: no algorithm takes a triangulated closed 4-manifold and returns its trisection genus.
• Takahashi (2023, revised 2025) found an exotic pair of 4-manifolds with boundary whose trisection genera are both 4, consistent with the relative version of the conjecture.

What would count as progress

• A proof of additivity in special cases, for example for connected sums in which one summand has small trisection genus, or within a restricted class of 4-manifolds.
• Computable lower bounds for trisection genus (for instance from the intersection form or Heegaard Floer invariants) that behave additively and are sharp on many examples.
• A counterexample, which would be a pair $X, Y$ with an explicit trisection diagram of $X \# Y$ of genus less than $g(X) + g(Y)$, together with lower bounds on $g(X)$ and $g(Y)$; trisection diagrams can be checked combinatorially.

References

1. D. T. Gay and R. Kirby, Trisecting 4-manifolds, Geom. Topol. 20 (2016), 3097–3132; arXiv:1205.1565.
2. J. Meier and A. Zupan, Genus-two trisections are standard, Geom. Topol. 21 (2017), 1583–1630.
3. P. Lambert-Cole and J. Meier, Bridge trisections in rational surfaces, J. Topol. Anal. 14 (2022), 655–708.
4. N. Takahashi, Exotic 4-manifolds with small trisection genus, arXiv:2308.00482 (2023).
5. N. M. Dunfield, M. Kegel, S. Y. Li and Q. Ren, Trisection invariants of 4-manifolds are uncomputable, arXiv:2608.27811 (2026).

Source

Imported from AIM Problem Lists trisections 1.1, curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.