Trisection genus

Revision as of 08:45, 31 August 2020

1 Problem

Let $== Problem == <wikitex>; 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. </wikitex> == References == {{#RefList:}} <!--[[Category:Questions]]--> [[Category:Research questions]] [[Category:Problems]]M$ be a closed smooth $4$-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

1. The trisection genus of $M$ is a homeomorphism invariant.
2. 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.

2 References