# Trisection genus

## 1 Problem

Let $M$$== Problem == ; Let M be a closed smooth -manifold. The ''trisection genus'' of M is the minimal genus of the central surface appearing in a trisection of M. '''Question:''' Is the trisection genus additive under connected sum? If so, then the following hold # The trisection genus of M is a homeomorphism invariant. # The manifolds S^4, \C P^2, S^2 \times S^2, \C P^2 \sharp \C P^2 and \C P^2 \sharp \overline{\C P^2} have a unique smooth structure. == References == {{#RefList:}} [[Category:Research questions]] [[Category:Problems]]M$ be a closed smooth $4$$4$-manifold. The trisection genus of $M$$M$ is the minimal genus of the central surface appearing in a trisection of $M$$M$.

Question: Is the trisection genus additive under connected sum?

If so, then the following hold

1. The trisection genus of $M$$M$ is a homeomorphism invariant.
2. The manifolds $S^4$$S^4$, $\C P^2$$\C P^2$, $S^2 \times S^2$$S^2 \times S^2$, $\C P^2 \sharp \C P^2$$\C P^2 \sharp \C P^2$ and $\C P^2 \sharp \overline{\C P^2}$$\C P^2 \sharp \overline{\C P^2}$ have a unique smooth structure.