arXiv research

Study trisections of non-orientable 4-manifolds with boundary.

New method distinguishes 4-manifold types using trisections.

New exotic 4-manifolds found with small trisection genus.

Two non-diffeomorphic minimal genus trisections found for a 4-manifold.

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the clo…

This paper proves a conjecture about trisections with a specific length.

New method for simplifying complex 4D shapes with boundaries.

Standard trisection diagrams found for Mazur type 4-manifolds.

Paper calculates homology and intersection form of trisected 4-manifolds with boundary.

The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.

Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, a…

The paper extends trisection theory to non-orientable 4-manifolds using colored triangulations.

Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms…

This paper constructs a Weinstein trisection for a surface bundle.

Gem theory helps estimate trisection genus of 4-manifolds.

Explains Kirby's work on 4-manifold trisections.

We study trisections of 4-manifolds obtained by spinning and twist-spinning 3-manifolds, and we show that, given a (suitable) Heegaard diagram for the 3-manifold, one can perform simple local modifications to obtain a trisection diagram for the 4-manifold. We also show that this local modification can be used to conver…

We classify a large class of "unbalanced" 4-manifold GK-trisections, which are a slight generalization of 4-manifold trisections defined by Gay and Kirby.

New trisection concept for non-closed 4-manifolds connects to group theory.

We develop a technique for gluing relative trisection diagrams of $4$-manifolds with nonempty connected boundary to obtain trisection diagrams for closed $4$-manifolds. As an application, we describe a trisection of any closed $4$-manifold which admits a Lefschetz fibration over $S^2$ equipped with a section of square …

The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every $k \geq 2$, we construct $2^{k}-1$ non-diffeomorphic $(3k,k)$-trisections on infinitely many 4-manifolds. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier's spun trisection…

Review of gem theory's interactions with Kirby diagrams and trisections.

The idea of studying trisections of closed smooth $4$-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and Tillmann to colored triangulations associated to the so called simple crystallization…

Symplectic 4-manifolds can be divided into three parts with a special structure.

New method constructs symplectic structures on 4-manifolds from trisections.

New 4-manifold invariant defined from trisection diagrams.

Notes describe geometric interpretations of cohomology in trisected 4-manifolds.

We extend the theory of relative trisections of smooth, compact, oriented $4$-manifolds with connected boundary given by Gay and Kirby to include $4$-manifolds with an arbitrary number of boundary components. Additionally, we provide sufficient conditions under which relatively trisected $4$-manifolds can be glued to o…

Study relates trisected 4-manifolds to cork twists via γ-curves.

We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized …

The paper extends trisection construction for Lefschetz fibrations with $(-n)$-sections.

Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.

Trisections are obtained by regluing surface-knots in 4-manifolds.

Paper proves a symplectic inequality using trisections and contact geometry.

The paper introduces a new complexity measure for 4-manifolds and connects it to the trisection genus.

We show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of $\natural^k (S^1 \times B^3)$, intersecting pairwise in 3--dimensional handlebodies, with triple intersection a closed 2--dimensional surface. Such a trisection is unique up to a natural stabilization operation. This …

Algorithm trisects 4D book into three chapters.

New invariant calculates 4-manifolds using trisection diagrams and combings.

In this paper, we introduce the relative $\mathcal{L}$-invariant $r\mathcal{L}(X)$ of a smooth, orientable, compact 4-manifold $X$ with boundary. This invariant is defined by measuring the lengths of certain paths in the cut complex of a trisection surface for $X$. This is motivated by the definition of the $\mathcal{L…

New proof confirms 4-manifolds with weakly reducible genus-three trisections are standard.

A simplified trisection is a trisection map on a 4-manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a 3-tuple of systems of simple closed curves in a surface to be a diagram of a simplifie…

We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3-sphere can be obtained from the standard Heegaard decomposition of $S^3$ by regluing…

The spine of a trisected 4-manifold is a singular 3-dimensional set from which the trisection itself can be reconstructed. 3-manifolds embedded in the trisected 4--manifold can often be isotoped to lie almost or entirely in the spine of the trisection. We define this notion and show that in fact every 3-manifold can be…

A theorem of Katanaga, Saeki, Teragaito, and Yamada relates Gluck and Price twists of 4-manifolds. Using trisection diagrams, we give a purely diagrammatic proof of this theorem, and answer a question of Kim and Miller.

Gay and Kirby introduced the notion of a trisection of a smooth 4-manifold, which is a decomposition of the 4-manifold into three elementary pieces. Rubinstein and Tillmann later extended this idea to construct multisections of piecewise-linear (PL) manifolds in all dimensions. Given a PL manifold $Y$ of dimension $n$,…

We present explicit algorithms for simplifying the topology of indefinite fibrations on 4-manifolds, which include broken Lefschetz fibrations and indefinite Morse 2-functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1-parameter families. In particular, given an arbitr…

Study on nonorientable 4-manifolds using simplified fibrations and trisections.