[Paper Review] Reflections on trisection genus

Michelle Chu, Stephan Tillmann|arXiv (Cornell University)|Sep 13, 2018

Geometric and Algebraic Topology2 references4 citations

TL;DR

This paper establishes new upper and lower bounds for the trisection genus of smooth, orientable 4-manifolds, particularly hyperbolic ones, using topological invariants and geometric constructions. It proves that trisection genus grows linearly with degree in finite covers and provides a refined upper bound of 7,201 and a lower bound of 96 for the Davis manifold, significantly improving prior estimates via tricolored triangulations and bistellar moves.

ABSTRACT

The Heegaard genus of a 3-manifold, as well as the growth of Heegaard genus in its finite sheeted cover spaces, has extensively been studied in terms of algebraic, geometric and topological properties of the 3-manifold. This note shows that analogous results concerning the trisection genus of a smooth, orientable 4-manifold have more general answers than their counterparts for 3-manifolds. In the case of hyperbolic 4-manifolds, upper and lower bounds are given in terms of volume and a trisection of the Davis manifold is described.

Motivation & Objective

• To establish tighter upper and lower bounds on trisection genus for smooth, orientable 4-manifolds, especially hyperbolic ones.
• To investigate how trisection genus behaves under finite-sheeted covers, particularly in relation to degree and Euler characteristic.
• To improve existing bounds on trisection genus using geometric and combinatorial techniques on Coxeter-type manifolds.
• To explore whether geometric invariants of hyperbolic 4-manifolds can yield stronger lower bounds on trisection genus.

Proposed method

• Derives a general lower bound of $ \frac{1}{3}|\chi(M)| $ for trisection genus when $ M \not\simeq S^4 $, using Euler characteristic and genus relations.
• Applies the upper bound $ g(M) \leq 60\sigma(M) $, where $ \sigma(M) $ is triangulation complexity, to estimate trisection genus.
• Uses a tricolored vertex partition of a 120-cell triangulation to define a piecewise-linear map to a 4-simplex, enabling trisection construction.
• Applies 2–4 bistellar moves to double-4-simplices in the triangulation to refine the decomposition into a trisection.
• Computes the genus of the central surface via counting squares and triangles in the trisection, yielding $ g(\Sigma) = 7,201 $ for the Davis manifold.
• Leverages the Coxeter group action on the 120-cell to define canonical graphs $ \Gamma_k $, which determine the topology of the 1–handlebodies in the trisection.

Experimental results

Research questions

• RQ1 For any finitely presented group $ G $, does there exist a 4-manifold $ M $ with $ \pi_1(M) = G $ and $ g(M) = \chi(M) - 2 + 3\operatorname{rk}(G) $?
• RQ2 How does trisection genus grow in finite-sheeted covers of 4-manifolds, particularly when $ \chi(M) \neq 0 $?
• RQ3 Can geometric invariants of hyperbolic 4-manifolds yield stronger lower bounds on trisection genus than algebraic or topological invariants alone?
• RQ4 What is the minimal possible trisection genus for the Davis manifold, and can it be further reduced using optimized triangulations?

Key findings

• The trisection genus of the Davis manifold satisfies $ 96 \leq g(M_D) \leq 7,201 $, improving upon the initial bound of $ 60 \cdot (120)^2 = 864,000 $.
• A refined construction using tricolored triangulation and 2–4 bistellar moves reduces the upper bound to 7,201, with the central surface genus computed as $ g(\Sigma) = 7,201 $.
• For any finite cover $ N \to M $ of degree $ d $, the trisection genus satisfies $ g(N) = \Theta(d) $ when $ \chi(M) \neq 0 $, and $ g(N) \geq \frac{1}{3}|\chi(M)|d $.
• The lower bound $ g(M) \geq \frac{1}{3}|\chi(M)| $ holds for all smooth, orientable 4-manifolds not diffeomorphic to $ S^4 $.
• The construction generalizes to other Coxeter-type hyperbolic 4-manifolds, suggesting potential for further improvements via optimized triangulations.
• The result confirms that trisection genus grows linearly with cover degree, and that triangulation complexity provides a useful proxy for trisection genus.