Vertical 3-manifolds in simplified genus-2 trisections of 4-manifolds

Abstract

We classify the $3$-manifolds obtained as the preimages of arcs on the plane for simplified genus-$2$ trisection maps, which we call vertical $3$-manifolds. Vertical $3$-manifolds of $(2, 1)$- and $(2, 2)$-trisection maps can be easily classified. A vertical $3$-manifold of a $(2, 0)$-trisection map is a connected sum of finite copies of $6$-tuple of vertical $3$-manifolds over specific $6$ arcs and $S^1 \times S^2$. Consequently, we show that each of the $6$-tuples determines the source $4$-manifold uniquely up to orientation reversing diffeomorphisms except for the trivial case. We also show that, in contrast to the fact that summands of vertical $3$-manifolds of simplified $(2, 0)$-trisection maps are lens spaces, there exist infinitely many simplified $(2, 0)$-$4$-section maps that admit hyperbolic vertical $3$-manifolds.