Is trisection genus additive under connected sum?

Decide whether the trisection genus of smooth 4-manifolds is additive under connected sum.

shoal State the problem, from AIM Problem Lists 33c5fd0 · 2周前历史

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1	# Is trisection genus additive under connected sum?
2
3	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.
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5	## Problem
6
7	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 $X$.
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9	Question. Is trisection genus additive under connected sum, that is,
10	$$g(X \# Y) = g(X) + g(Y)$$
11	for all such 4-manifolds $X$ and $Y$?
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13	The inequality $g(X \# Y) \le g(X) + g(Y)$ is easy, by gluing trisections; the question is whether genus can drop.
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15	## Background
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17	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).
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19	## What is known
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21	- Trisections exist and are unique up to stabilisation (Gay and Kirby, 2016).
22	- 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.
23	- Lambert-Cole and Meier (2022) conjectured additivity and showed that it implies equal trisection genus for exotic pairs of closed 4-manifolds.
24	- 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.
25	- 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.
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27	## What would count as progress
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29	- 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.
30	- 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.
31	- 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.
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33	## References
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35	1. D. T. Gay and R. Kirby, [Trisecting 4-manifolds](https://doi.org/10.2140/gt.2016.20.3097), Geom. Topol. 20 (2016), 3097–3132; [arXiv:1205.1565](https://arxiv.org/abs/1205.1565).
36	2. J. Meier and A. Zupan, [Genus-two trisections are standard](https://doi.org/10.2140/gt.2017.21.1583), Geom. Topol. 21 (2017), 1583–1630.
37	3. P. Lambert-Cole and J. Meier, [Bridge trisections in rational surfaces](https://doi.org/10.1142/S1793525321500047), J. Topol. Anal. 14 (2022), 655–708.
38	4. N. Takahashi, [Exotic 4-manifolds with small trisection genus](https://arxiv.org/abs/2308.00482), arXiv:2308.00482 (2023).
39	5. N. M. Dunfield, M. Kegel, S. Y. Li and Q. Ren, [Trisection invariants of 4-manifolds are uncomputable](https://arxiv.org/abs/2608.27811), arXiv:2608.27811 (2026).
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41	## Source
42
43	Imported from [AIM Problem Lists trisections 1.1](http://aimpl.org/trisections/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.