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Distinguishing diffeomorphism types of relative trisections

Natsuya Takahashi

Abstract

We distinguish diffeomorphism types of relative trisections using a ``capping'' operation, which yields a trisection diagram of a closed 4-manifold from a relative trisection diagram. Using this operation, we give various examples of non-diffeomorphic relative trisections of the same 4-manifold with boundary. We also study how the corresponding trisected 4-manifold changes under the capping operation.

Distinguishing diffeomorphism types of relative trisections

Abstract

We distinguish diffeomorphism types of relative trisections using a ``capping'' operation, which yields a trisection diagram of a closed 4-manifold from a relative trisection diagram. Using this operation, we give various examples of non-diffeomorphic relative trisections of the same 4-manifold with boundary. We also study how the corresponding trisected 4-manifold changes under the capping operation.
Paper Structure (12 sections, 20 theorems, 31 equations, 31 figures)

This paper contains 12 sections, 20 theorems, 31 equations, 31 figures.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Trisections of closed $4$-manifolds
  4. Relative trisections of $4$-manifolds with boundary
  5. Trisection diagram
  6. Distinguishing relative trisections by the capping operation
  7. Proofs of Theorems \ref{['thm:cl+1hdlbd']} and \ref{['thm:-D4']}
  8. Proof of Theorem \ref{['thm:ndrt-inf']}
  9. Proof of Theorem \ref{['thm:ndrt-Ac']}
  10. Proof of Theorem \ref{['thm:non-min-genus']}
  11. A geometric meaning of the capping operation
  12. Minimal genus relative trisections with distinct parameters

Key Result

Theorem 1.2

If $\mathcal{D}$ is a $(g,k;0,b)$-relative trisection diagram, then $\mathrm{Cap}\left( \mathcal{D} \right)$ is a $(g,k-b+1)$-trisection diagram of a closed $4$-manifold. In addition, if two $(g,k;0,b)$-relative trisection diagrams $\mathcal{D}$ and $\mathcal{D}'$ are diffeomorphism and handleslide

Figures (31)

  • Figure 1: Left: relative trisection diagram $\mathcal{D}=(\Sigma;\alpha,\beta,\gamma)$. Right: trisection diagram $\mathrm{Cap}\left( \mathcal{D} \right)=(\mathrm{Cap}\left( \Sigma \right);\alpha,\beta,\gamma)$.
  • Figure 2: The boundary connected sum $Z_{k}={U_{p,b}} \,\natural\, {V_{n}}$.
  • Figure 3: The standard diagram $(\Sigma_{g,b};\delta,\epsilon)$.
  • Figure 4: The standard diagram $(\Sigma_{g,b};\delta,\epsilon)$.
  • Figure 5: $(2,1;0,2)$-relative trisection diagrams of $S^2\times D^2$.
  • ...and 26 more figures

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem 1.2: Tak24
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 30 more