The paper generalizes trisection definitions and applies them to specific cases.
problem Defining and applying trisections to 4-manifolds and surfaces.
method Generalized trisection definitions and their application to specific cases.
result Classification of an infinite family of genus three trisections.
New trisection concept for non-closed 4-manifolds connects to group theory.
problem Defining and analyzing trisections of non-closed 4-manifolds.
method Generalized group trisections to non-closed cases and established a one-to-one correspondence.
result Established a functorial relationship between relative trisections and groups.
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…
Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.
problem Creating efficient trisection diagrams for 4-manifolds.
method Algorithm converting Kirby diagrams to trisection diagrams.
result Provides examples of trisection diagrams for 4-manifolds.
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 S2 equipped with a section of square …
Paper studies simplified trisections and their equivalence classes.
problem Understanding right-left equivalence of simplified (2,0)-trisections. method Analyzes simplified trisection diagrams and upper-triangular handle-slides.
result At least two simplified (2,0)-trisections can be right-left equivalent without being related by automorphisms or handle-slides. New method distinguishes 4-manifold types using trisections.
problem Distinguishing different 4-manifold types.
method Capping operation to transform relative trisections into closed 4-manifold diagrams.
result Examples of non-diffeomorphic relative trisections of the same 4-manifold.
Gem theory helps estimate trisection genus of 4-manifolds.
problem Estimating the trisection genus of 4-manifolds.
method Using gem theory, a type of edge-colored graphs dual to colored triangulations.
result Regular genus is an upper bound for trisection genus of closed 4-manifolds.
This paper constructs a Weinstein trisection for a surface bundle.
problem Constructing a trisection for a surface bundle.
method Weinstein trisection approach applied to ΣgimesΣh. result Explicit construction of a Weinstein trisection for S2imesS2. The paper extends trisection construction for Lefschetz fibrations with (−n)-sections.
problem Constructing trisections for 4-manifolds with specific Lefschetz fibrations.
method Explicit construction of trisections using monodromies of Lefschetz fibrations.
result Trisections for various 4-manifolds constructed from Lefschetz fibrations.
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 constructs trisections for fiber bundles over a circle.
problem Building trisections for fiber bundles over the circle.
method Using sutured Heegaard splittings and diagrams, the paper provides an algorithm to construct relative trisection diagrams.
result Explicit construction of relative trisection diagrams for fiber bundles over the circle.
Study trisections of non-orientable 4-manifolds with boundary.
problem Understanding trisections in non-orientable 4-manifolds.
method Introduced trisections of non-orientable 4-manifolds with boundary, proved a non-orientable analogue of a theorem, and discussed adaptation of trisection theory.
result Existence of trisection diagrams and Kirby diagrams for closed non-orientable 4-manifolds.
The paper extends trisection theory to non-orientable 4-manifolds using colored triangulations.
problem Extending trisection theory to non-orientable 4-manifolds.
method Using colored triangulations and gems to induce trisections.
result Gem-induced trisections naturally give rise to trisections of closed 4-manifolds.
Two non-diffeomorphic minimal genus trisections found for a 4-manifold.
problem Existence of non-diffeomorphic minimal genus trisections of the same 4-manifold.
method Introduced a simple operation to create a trisection diagram from a relative trisection diagram.
result Existence of two non-diffeomorphic minimal genus trisections of the same (g,k;p,b)-type 4-manifold. Paper calculates homology and intersection form of trisected 4-manifolds with boundary.
problem Calculating homology and intersection form of trisected 4-manifolds with boundary.
method Uses relative trisection diagrams to calculate homology and intersection form.
result Describes a representative of the second Stiefel-Whitney class using relative trisection diagrams.
We define a trisection of a closed, orientable three dimensional manifold into three handlebodies, and a notion of stabilization for these trisections. Several examples of trisections are described in detail. We define the trisection genus t(M) of a 3-manifold, and relate it to the Heegaard genus g(M), showing that…
The paper generalizes trisections to compact PL 4-manifolds with boundary and introduces gem-induced trisections.
problem Generalizing trisections to compact PL 4-manifolds with boundary and any 5-colored graph encoding simply-connected 4-manifolds.
method Introducing gem-induced trisections and analyzing their properties for compact PL 4-manifolds with and without boundary.
result Conditions for gem-induced trisections to realize the G-trisection genus and direct determination of the genus from the graph.
We show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of ♮k(S1×B3), 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 …
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 summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of t…
New method constructs symplectic structures on 4-manifolds from trisections.
problem Characterize 4-manifolds admitting symplectic structures.
method Explicit criteria on trisections allow construction of symplectic structures.
result New characterization of 4-manifolds with symplectic structures.
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…
We generalize bridge trisections to surfaces in four-manifolds, linking them to braided links.
problem Embedding surfaces in four-manifolds and understanding their braiding.
method Introducing bridge trisections, isotoping surfaces, and using trisections and open-book decompositions.
result Any neatly embedded surface can be isotoped to lie in bridge trisection position with respect to any trisection of the four-manifold.
Donaldson showed that every closed symplectic 4-manifold can be given the structure of a topological Lefschetz pencil. Gay and Kirby showed that every closed 4-manifold has a trisection. In this paper we relate these two structure theorems, showing how to construct a trisection directly from a topological Lefschetz pen…
Trisections are obtained by regluing surface-knots in 4-manifolds.
problem Understanding trisections in 4-manifolds using surface-knots.
method Trivial gluing of relative trisections of ν(S) and X−ν(S). result Trisection obtained is diffeomorphic to a stabilization of the original trisection.
Multisections generalize Heegaard splittings and trisections to higher dimensions.
problem Decomposing higher-dimensional manifolds into 1-handlebodies with specific intersection properties.
method Considering multisections of higher-dimensional smooth or PL closed orientable manifolds, and associating diagrams to them.
result Multisection diagrams uniquely determine manifolds in all dimensions up to 6.
Equivariant trisections for group actions on 4-manifolds are introduced and studied.
problem Understanding the equivariant topology of G-manifolds and their quotients. method Introducing G-equivariant trisections and bridge trisections, and establishing their existence for G-manifolds. result Any G-manifold X admits a G-equivariant trisection such that a G-invariant surface S is in equivariant bridge trisection position. The paper introduces a new complexity measure for 4-manifolds and connects it to the trisection genus.
problem Defining and analyzing a new complexity measure for 4-manifolds.
method Defining a new complexity measure scr and proving an inequality involving the trisection genus. result Proves an inequality relating the trisection genus to the new complexity measure.
Review of gem theory's interactions with Kirby diagrams and trisections.
problem Representing and understanding PL 4-manifolds.
method Combining gem theory with Kirby diagrams and trisections.
result New gems representing 4-manifolds requiring 3-handles.
Study relates trisected 4-manifolds to cork twists via γ-curves.
problem Understanding relationships between trisected 4-manifolds and cork twists.
method Analyzes γ-curves derived from topology of relative trisected 4-manifolds and their connections to Torelli and Johnson kernels.
result Establishes a connection between γ-curves and cork twists in relative trisected 4-manifolds.
Algorithm trisects 4D book into three chapters.
problem Trisecting 4D book into chapters.
method Algorithm for open book decompositions of 4-manifolds.
result Algorithm works for various types of manifolds.
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…
We establish a correspondence between trisections of smooth, compact, oriented 4--manifolds with connected boundary and diagrams describing these trisected 4--manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fe…
Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold X as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface Σ, guiding the gluing of the handlebodies. Any morphism φ from …
We show that the only closed 4-manifolds admitting genus two trisections are S2×S2 and connected sums of S1×S3, CP2, and CP2 with two summands. Moreover, each of these manifolds admits a unique genus two trisection up to diffeomorphism. The proof relies heavily o…
Paper proves a symplectic inequality using trisections and contact geometry.
problem Proving a symplectic inequality for 4-manifolds.
method Used contact geometry and trisections to reduce to slice-Bennequin inequality, proving with Khovanov homology.
result Gauge-theory-free proofs of landmark 4-manifold topology results.
In this paper, we introduce the relative L-invariant rL(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…
In 2018, M. Chu and S. Tillmann gave a lower bound for the trisection genus of a closed 4-manifold in terms of the Euler characteristic of M and the rank of its fundamental group. We show that given a group G, there exist a 4-manifold M with fundamental group G with trisection genus achieving Chu-Tillmann's low…
Multisections generalize trisections for 4-manifolds, allowing complex operations and explicit diagrams.
problem Decomposing arbitrary smooth 4-manifolds into simpler pieces.
method Introducing multisections as a sequence of cut systems on a surface, enabling smooth operations and explicit diagrams.
result Elliptic fibrations admit genus 3 multisections, with explicit diagrams.
Study on nonorientable 4-manifolds using simplified fibrations and trisections.
problem Classify and understand nonorientable 4-manifolds.
method Use simplified broken Lefschetz fibrations and trisections, topological modifications of singularities, handlebody decompositions, and mapping classes of surfaces.
result Classify low genus simplified broken Lefschetz fibrations on nonorientable 4-manifolds.
Recently Gay and Kirby described a new decomposition of smooth closed 4-manifolds called a trisection. This paper generalises Heegaard splittings of 3-manifolds and trisections of 4-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear n-mani…
Algorithms compute invariants of 4-manifolds as branched covers.
problem Computing invariants of 4-manifolds as branched covers.
method Diagrammatic algorithms using tri-plane diagrams and permutations.
result Automated algorithm for computing Kjuchukova's homotopy-ribbon obstruction.
Given two smooth, oriented, closed 4-manifolds M1 and M2, we construct two invariants, DP(M1,M2) and D(M1,M2), coming from distances in the pants complex and the dual curve complex respectively. To do this, we adapt work of Johnson on Heegaard splittings of 3-manifolds to the trisections of 4-manifolds…
This paper proves a conjecture about trisections with a specific length.
problem Proving a conjecture about trisections with a specific length.
method Examining trisections with Kirby-Thompson length 2 and proving the conjecture.
result Proves the conjecture about length 2 trisection being a 4-manifold with length 0.
The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.
problem Determining the trisection genus of contractible 4-manifolds and exotic pairs.
method Proved trisection genus of Akbulut cork and constructed corks with trisection genus 3.
result First examples of contractible 4-manifolds with determined trisection genus except for 4-ball.
New method for simplifying complex 4D shapes with boundaries.
problem Complexity reduction in 4D shape analysis.
method Introducing pseudo-trisections and pseudo-bridge trisections.
result Existence and uniqueness of pseudo-trisections established.
Classifies knot traces with specific trisection genus limits.
problem Classifying knot traces with specific trisection genus limits.
method Classifying knot traces with specific trisection genus limits.
result Infinitely many knots have traces with trisection genus 3 and 4, and arbitrarily large trisection genus.