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Exotic 4-manifolds with small trisection genus

Natsuya Takahashi

TL;DR

The paper investigates how small relative trisection genus can reflect exotic smooth structures on 4-manifolds with boundary. It develops explicit genus-$3$ relative trisections for Mazur-type contractible manifolds $W^ obreaker^ obreaker(l,k)$ and constructs a low-genus exotic pair $(P_1,Q_1)$ with $g(P_1)=g(Q_1)=4$, using cork twists along the Akbulut cork. The constructions rely on detailed relative trisection diagrams and a realization algorithm (Kim–Miller), connecting cork theory to 4-manifold trisections. The results provide evidence for conjectures on additivity and exotic-tg invariance and raise questions about the existence of even smaller genus examples and the precise genus for certain Mazur-type manifolds.

Abstract

We show that there exists an exotic pair of $4$-manifolds with boundary whose trisection genera are $4$. We also construct genus-$3$ relative trisections for an infinite family of contractible $4$-manifolds introduced by Akbulut and Kirby.

Exotic 4-manifolds with small trisection genus

TL;DR

The paper investigates how small relative trisection genus can reflect exotic smooth structures on 4-manifolds with boundary. It develops explicit genus- relative trisections for Mazur-type contractible manifolds and constructs a low-genus exotic pair with , using cork twists along the Akbulut cork. The constructions rely on detailed relative trisection diagrams and a realization algorithm (Kim–Miller), connecting cork theory to 4-manifold trisections. The results provide evidence for conjectures on additivity and exotic-tg invariance and raise questions about the existence of even smaller genus examples and the precise genus for certain Mazur-type manifolds.

Abstract

We show that there exists an exotic pair of -manifolds with boundary whose trisection genera are . We also construct genus- relative trisections for an infinite family of contractible -manifolds introduced by Akbulut and Kirby.
Paper Structure (6 sections, 7 theorems, 1 equation, 19 figures)

This paper contains 6 sections, 7 theorems, 1 equation, 19 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relative trisections
  4. Corks and Mazur-type $4$-manifolds
  5. Relative trisections of $W^\pm(l,k)$.
  6. Trisection genera of exotic $4$-manifolds with boundary.

Key Result

Theorem 1.2

There exists an exotic pair $(P_1, Q_1)$ of $4$-manifolds with boundary such that $g(P_1)=g(Q_1)=4$.

Figures (19)

  • Figure 1:
  • Figure 3:
  • Figure 5: The standard diagram $(\Sigma;\delta,\epsilon)$ of type $(g,k;p,b)$.
  • Figure 6:
  • Figure 8: Left: $\mathcal{D}_n$ for $n>0$. Right: $\mathcal{D}_n$ for $n\leq0$.
  • ...and 14 more figures

Theorems & Definitions (21)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Castro--Gay--Pinzón-Caiced CasGayPin18_1
  • Definition 2.5
  • Example 2.6: Akbulut--Yasui AkbYas13, see also Akb91_1 for the case $n=1$
  • Proposition 2.7: Akbulut--Kirby AkbKir79, see also Akb16a
  • ...and 11 more