arXiv subjects
Publications and source records attributed to Nobutaka Asano.
We study Stein fillings of planar contact 3-manifolds under certain constraints on their relative trisections. We partially classify the diffeomorphism types of such fillings admitting relative trisections with genus at most 2.
Kirby and Thompson introduced a non-negative integer-valued invariant, called the Kirby-Thompson invariant, of a $4$-manifold using trisections. In this paper, we give some lower bounds for the Kirby-Thompson invariant of certain $4$-manifolds. As an application, we determine the Kirby-Thompson invariant of the spin of $L(2,1)$, which is the first example of a $4$-manifold with non-trivial Kirby-Thompson invariant. We also show that there exist $4$-manifolds with arbitrarily large Kirby-Thompson invariant.
Trisection maps are certain stable maps from closed $4$--manifolds to $\mathbf{R}^2$. A simpler but reasonable class of trisection maps was introduced by Baykur and Saeki, called a simplified $(g, k)$-trisection. We focus on the right-left equivalence classes of simplified $(2, 0)$-trisections. Simplified trisections are determined by their simplified trisection diagrams, which are diagrams on a genus-$2$ surface consisting of simple closed curves of vanishig cycles with labels. The aim of this paper is to study how the replacement of reference paths changes simplified trisection diagrams up to upper-triangular handle-slides. We show that for a simplified trisection $f$ satisfying a certain condition, there exists at least two simplified $(2, 0)$-trisections $f'$ and $f''$ such that $f, f'$ and $f''$ are right-left equivalent to each other but their simplified trisection diagrams are not related by automorphisms of a genus-$2$ surface and upper-triangular handle-slides.
We classify the $3$-manifolds obtained as the preimages of arcs on the plane for simplified $(2, 0)$-trisection maps, which we call vertical $3$-manifolds. Such a $3$-manifold is a connected sum of a $6$-tuple of vertical $3$-manifolds over specific $6$ arcs. Consequently, we show that each of the $6$-tuples determines the source $4$-manifold uniquely up to orientation reversing diffeomorphisms. 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.