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.
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.
New exotic 4-manifolds found with small trisection genus.
problem Finding exotic 4-manifolds with small trisection genus.
method Constructing genus-3 relative trisections for contractible 4-manifolds.
result Exotic 4-manifolds with boundary have trisection genus of 4.
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. 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.
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.
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.
Standard trisection diagrams found for Mazur type 4-manifolds.
problem Finding standard trisection diagrams for Mazur type 4-manifolds.
method Doubling a relative trisection diagram and using an algorithm from Kirby diagrams to trisection diagrams.
result Certain trisection diagrams of Mazur type 4-manifolds are standard.
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.
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.
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.
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.
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.
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.
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 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.
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.
Explains Kirby's work on 4-manifold trisections.
problem Understanding trisections of 4-manifolds.
method Near-symplectic constructions.
result Developed a theory of 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…
New 4-manifolds meet trisection bounds.
problem Finding 4-manifolds with specific trisection genus.
method Constructing 4-manifolds with fundamental group G achieving Chu-Tillmann's lower bound. result Existence of 4-manifolds meeting trisection genus bounds.
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.
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.
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 …
The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every k≥2, we construct 2k−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.
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.
Diagrammatic proof of twist theorem for 4-manifolds.
problem Understanding twists of 4-manifolds.
method Using trisection diagrams for purely diagrammatic proof.
result Answered a question about twists of 4-manifolds.
Symplectic 4-manifolds can be divided into three parts with a special structure.
problem Understanding the structure of symplectic 4-manifolds.
method Proved the existence of a trisection compatible with the symplectic structure.
result Symplectic 4-manifolds admit a trisection compatible with the symplectic structure.
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.
New 4-manifold invariant defined from trisection diagrams.
problem Defining a new 4-manifold invariant from trisection diagrams.
method Algebraic data from bimodule categories and spherical fusion categories, described diagrammatically.
result Includes Hopf algebraic invariants and modular fusion category invariants.
3-manifolds can be embedded almost in the spine of a trisected 4-manifold.
problem Embedding 3-manifolds in trisected 4-manifolds.
method Defining and using the spine of a trisected 4-manifold, isotoping 3-manifolds to lie in the spine, and calculating bounds based on graph distances.
result Every 3-manifold can be embedded almost in the spine of a minimal genus trisection of connect sums of S2ildeimesS2. This note extends trisection genus results from 3-manifolds to 4-manifolds.
problem Understanding trisection genus in 4-manifolds.
method Analogous results to 3-manifolds, with bounds given in terms of volume and a specific trisection.
result Upper and lower bounds for trisection genus in hyperbolic 4-manifolds.
Study intersection forms of 4-manifolds using trisections and mapping class groups.
problem Understanding intersection forms of 4-manifolds via trisections and mapping class groups.
method Apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds.
result Analogous result for trisections of 4-manifolds analogous to Morita's for homology 3-spheres.
Notes describe geometric interpretations of cohomology in trisected 4-manifolds.
problem Understanding geometric interpretations of cohomology classes in trisected 4-manifolds.
method Analogy with Hodge theory and sheaf cohomology in algebraic geometry.
result Classes in H2(X) can be interpreted as (1,1)-classes. 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.
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.
4-manifolds with specific trisections are 3-fold covers of S^4.
problem Understanding 4-manifolds with (g;k1,k2,0)-trisections. method Proving 4-manifolds with certain trisections are irregular 3-fold covers of S4. result Any 4-manifold with a (g;k1,k2,0)-trisection is an irregular 3-fold cover of S4. 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 …
Paper defines a new invariant for 4-manifolds with boundary.
problem Defining a new invariant for 4-manifolds with boundary.
method Defined the relative L-invariant by measuring path lengths in the cut complex of a trisection surface.
result Shows that if a rational homology ball has a zero relative L-invariant, it must be the 4-ball.
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.
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.
Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
problem Classifying 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
method Classifies vertical 3-manifolds as preimages of arcs on the plane for simplified (2,0)-trisection maps.
result Each 6-tuple of vertical 3-manifolds determines the source 4-manifold uniquely up to orientation reversing diffeomorphisms.
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.
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.
The abstract discusses how trisections relate to 4-manifolds, similar to how Heegaard splittings relate to 3-manifolds.
problem Exploring the analogy between Heegaard splittings and trisections in 3D and 4D manifolds.
method Develops the analogy through definitions, Morse theory, and diagrammatic descriptions in both 4-manifolds and those with boundaries or embedded submanifolds.
result Clarifies the distinction between different perspectives on decompositions of manifolds in 4D.
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.
This research connects 4-manifold trisections to their underlying surfaces and their intersection forms.
problem Understanding the intersection forms of 4-manifolds from their trisection diagrams.
method Expressing the twisted homology and Reidemeister torsion of a 4-manifold in terms of the first homology of its trisection surface and the intersection form.
result The intersection form of a 4-manifold can be derived from the intersection form of its trisection surface.
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 …
New invariant calculates 4-manifolds using trisection diagrams and combings.
problem Calculating non-semisimple 4-manifold invariants.
method Using trisection diagrams and combings of the trisection surface.
result Invariant calculated for Stein nuclei, generalizing earlier semisimple version.