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Estimating trisection genus via gem theory

Maria Rita Casali, Paola Cristofori

TL;DR

The paper links gem theory with 4-manifold trisections by showing that the regular genus of a gem-based representation provides an upper bound for trisection genus, and by giving a concrete procedure to derive trisection diagrams directly from gems. It generalizes trisection construction to compact 4-manifolds with connected boundary via gem-induced trisections and stabilizations along 4-colored edges, yielding quantitative bounds: $g_T(bar M) \le g_{GT}(M) \le G(M)$ (and related bounds involving boundary Heegaard genus and fundamental-group rank). It also provides explicit recipes to obtain trisection diagrams, and G-trisection diagrams for simply-connected cases, directly from a given gem. These results integrate combinatorial gem theory with 4-manifold topology, enabling more efficient estimations of trisection genus and direct diagrammatic representations from colored graphs.

Abstract

Gems are a particular type of edge-colored graphs, dual to colored triangulations, which represent compact PL-manifolds of arbitrary dimension, both in the closed and boundary case. In the present paper, gem theory is used to approach trisections of PL 4-manifolds, so as to prove that: - the graph-defined invariant regular genus is an upper bound for the trisection genus of each closed 4-manifold; - a trisection diagram can be directly obtained from any gem of a closed 4-manifold. Moreover, suitable extensions of the above results are presented for compact 4-manifolds with connected boundary.

Estimating trisection genus via gem theory

TL;DR

The paper links gem theory with 4-manifold trisections by showing that the regular genus of a gem-based representation provides an upper bound for trisection genus, and by giving a concrete procedure to derive trisection diagrams directly from gems. It generalizes trisection construction to compact 4-manifolds with connected boundary via gem-induced trisections and stabilizations along 4-colored edges, yielding quantitative bounds: (and related bounds involving boundary Heegaard genus and fundamental-group rank). It also provides explicit recipes to obtain trisection diagrams, and G-trisection diagrams for simply-connected cases, directly from a given gem. These results integrate combinatorial gem theory with 4-manifold topology, enabling more efficient estimations of trisection genus and direct diagrammatic representations from colored graphs.

Abstract

Gems are a particular type of edge-colored graphs, dual to colored triangulations, which represent compact PL-manifolds of arbitrary dimension, both in the closed and boundary case. In the present paper, gem theory is used to approach trisections of PL 4-manifolds, so as to prove that: - the graph-defined invariant regular genus is an upper bound for the trisection genus of each closed 4-manifold; - a trisection diagram can be directly obtained from any gem of a closed 4-manifold. Moreover, suitable extensions of the above results are presented for compact 4-manifolds with connected boundary.

Paper Structure

This paper contains 5 sections, 11 theorems, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic notions of gem theory
  3. Trisections arising from gems
  4. Estimations of the trisection genus via gems
  5. From gems to trisection diagrams

Key Result

Theorem 1

Let $M$ be a compact $4$-manifold whose boundary is either empty or a connected sum of sphere bundles over $\mathbb S^1$; then: where $\mathcal{G}(M)$ denotes the regular genus of $M$, while $g_T(\bar{M})$ is the trisection genus of the closed 4-manifold $\bar{M}$ (uniquely) associated to $M$. In particular, if $M$ is closed, then:

Figures (3)

  • Figure 1: The intersections of the 3-dimensional pieces and the central surface of $\mathcal{T}(\Gamma, \varepsilon)$ with any 4-simplex (redrawn from Spreer-Tillmann(Exp))
  • Figure 2: the square, corresponding to a $4$-colored edge of $\Gamma$, constituting $Q(\Gamma,\varepsilon)$
  • Figure 3: stabilization along a $4$-colored edge

Theorems & Definitions (26)

  • Theorem 1
  • Definition 1
  • Remark 1
  • Theorem 2
  • Proposition 3
  • Definition 2
  • Remark 2
  • Definition 3
  • Remark 3
  • Remark 4
  • ...and 16 more