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Pseudo-trisections of four-manifolds with boundary

Shintaro Fushida-Hardy

TL;DR

The paper introduces pseudo-trisections as a lower-complexity alternative to relative trisections for compact 4-manifolds with boundary and proves their existence, uniqueness up to three stabilisation moves, and a diagrammatic correspondence. It extends the framework to pseudo-trisection diagrams and proves a realisation bijection up to diagrammatic moves, enabling a calculus that often yields simpler representations than relative trisections. The work further develops pseudo-bridge trisections and pseudo-shadow diagrams to study neatly embedded surfaces, including crossing-resolution rules and invariants computable from diagrams. The resulting diagrammatic theory encodes both ambient 4-manifolds with boundary and embedded surfaces, with applications demonstrated through explicit examples and invariant computations. Overall, pseudo-trisections provide a versatile, lower-complexity toolkit for 4-manifolds with boundary and embedded surfaces, with broad diagrammatic methods for analysis and computation.

Abstract

We introduce the concept of pseudo-trisections of smooth oriented compact 4-manifolds with boundary. The main feature of pseudo-trisections is that they have lower complexity than relative trisections for given 4-manifolds. We prove existence and uniqueness of pseudo-trisections, and further establish a one-to-one correspondence between pseudo-trisections and their diagrammatic representations. We next introduce the concept of pseudo-bridge trisections of neatly embedded surfaces in smooth oriented compact 4-manifolds. We develop a diagrammatic theory of pseudo-bridge trisections and provide examples of computations of invariants of neatly embedded surfaces in 4-manifolds using said diagrams.

Pseudo-trisections of four-manifolds with boundary

TL;DR

The paper introduces pseudo-trisections as a lower-complexity alternative to relative trisections for compact 4-manifolds with boundary and proves their existence, uniqueness up to three stabilisation moves, and a diagrammatic correspondence. It extends the framework to pseudo-trisection diagrams and proves a realisation bijection up to diagrammatic moves, enabling a calculus that often yields simpler representations than relative trisections. The work further develops pseudo-bridge trisections and pseudo-shadow diagrams to study neatly embedded surfaces, including crossing-resolution rules and invariants computable from diagrams. The resulting diagrammatic theory encodes both ambient 4-manifolds with boundary and embedded surfaces, with applications demonstrated through explicit examples and invariant computations. Overall, pseudo-trisections provide a versatile, lower-complexity toolkit for 4-manifolds with boundary and embedded surfaces, with broad diagrammatic methods for analysis and computation.

Abstract

We introduce the concept of pseudo-trisections of smooth oriented compact 4-manifolds with boundary. The main feature of pseudo-trisections is that they have lower complexity than relative trisections for given 4-manifolds. We prove existence and uniqueness of pseudo-trisections, and further establish a one-to-one correspondence between pseudo-trisections and their diagrammatic representations. We next introduce the concept of pseudo-bridge trisections of neatly embedded surfaces in smooth oriented compact 4-manifolds. We develop a diagrammatic theory of pseudo-bridge trisections and provide examples of computations of invariants of neatly embedded surfaces in 4-manifolds using said diagrams.
Paper Structure (22 sections, 28 theorems, 23 equations, 30 figures)

This paper contains 22 sections, 28 theorems, 23 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Trisections of 3-manifolds
  3. 3-manifold trisections and stabilisation
  4. Heegaard stabilisation
  5. Triple Heegaard diagrams
  6. Complexity of 3-manifold trisections
  7. Pseudo-trisections
  8. Pseudo-trisections and their properties
  9. Relative trisections
  10. Examples of pseudo-trisections
  11. Stabilisations and uniqueness of pseudo-trisections
  12. Complexity of pseudo-trisections
  13. Pseudo-trisection diagrams
  14. Pseudo-trisection diagrams
  15. Boundary stabilisation shift and relative existence
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $X$ be a compact oriented smooth 4-manifold with non-empty connected boundary $Y$. Let $\tau$ be a trisection of $Y$. Then there is a pseudo-trisection of $X$ which restricts to $\tau$ on the boundary.

Figures (30)

  • Figure 1: The components of a trisection of a 3-manifold $Y$.
  • Figure 2: A stabilisation of a trisection of a 3-manifold.
  • Figure 3: A Heegaard stabilisation of a trisection of a 3-manifold.
  • Figure 4: A triple Heegaard diagram of $T^3$.
  • Figure 5: Stabilisation of triple Heegaard diagrams.
  • ...and 25 more figures

Theorems & Definitions (120)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 110 more