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.
Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
New proof shows 4-manifolds can't support complex structures.
problem Proving 4-manifolds can't support complex structures.
method New proof of complex structure non-existence for real-hyperbolic 4-manifolds.
result Extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure.
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.
Study on lens spaces bounding 4-manifolds with specific Betti numbers.
problem Which lens spaces can bound 4-manifolds with second Betti number one?
method Construction of specific 4-manifolds and analysis of lens space boundaries.
result Infinite families of lens spaces can bound 4-manifolds with second Betti number one, but not all.
4-manifolds with non-positive curvature are essentially Euclidean.
problem Understanding the structure of 4-manifolds with specific curvature properties.
method Proving homeomorphism to Euclidean space using globally non-positive curvature.
result CAT(0) 4-manifolds are homeomorphic to Euclidean space.
Four hyperbolic 24-cell 4-manifolds with one cusp are identified.
problem Identifying hyperbolic 24-cell 4-manifolds with one cusp.
method Analyzing hyperbolic geometry and isometry.
result Found four one-cusped hyperbolic 4-manifolds of minimum volume.
Estimates intersection pairing in hyperbolic 4-manifolds.
problem Estimating intersection pairing in hyperbolic 4-manifolds.
method Using Thurston norms of homology classes.
result Proved an estimate on intersection pairing.
Study immersions of punctured 4-manifolds for quantum automata applications.
problem Existence of immersions between specific 4-manifolds.
method Analyzing immersions of punctured 4-manifolds to establish a partial order.
result Established a partial order on closed 4-manifolds via immersions.
New exotic 4-manifolds with zero signature found.
problem Finding exotic 4-manifolds with specific properties.
method Special small Lefschetz fibrations built via positive factorizations in the mapping class group.
result Smallest known exotic closed simply connected 4-manifolds with signature zero.
No Einstein metrics found on extended graph 4-manifolds.
problem Finding Einstein metrics on extended graph 4-manifolds.
method Defined and analyzed extended graph 4-manifolds as per [FLS15].
result Extended graph 4-manifolds do not support Einstein metrics.
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…
Study on spheres in simply-connected 4-manifolds with abelian complements.
problem Classifying locally flat spheres in simply-connected 4-manifolds.
method Focus on complements with abelian fundamental groups.
result New classifications of spheres in specific 4-manifolds.
New 4-manifolds with exotic diffeomorphisms found.
problem Existence of exotic diffeomorphisms in 4-manifolds.
method Proves existence of infinitely many contractible 4-manifolds with exotic diffeomorphisms.
result Infinitely many contractible 4-manifolds with absolutely exotic diffeomorphisms.
Study exotic Dehn twists in 4-manifolds, producing first known exotic diffeomorphisms.
problem Detecting exotic diffeomorphisms in 4-manifolds.
method 2-parameter families Seiberg-Witten theory over RP2. result Constructed the smallest closed 4-manifold with exotic diffeomorphisms.
The paper finds infinitely many 4-manifolds with non-isotopic sections.
problem Finding non-isotopic sections in 4-manifolds.
method Using Nielsen equivalence, the paper constructs infinitely many 4-manifolds with non-isotopic sections.
result Infinitely many 4-manifolds have non-isotopic sections.
Study on fundamental groups of framed circle embeddings in 4-manifolds.
problem Understanding the fundamental groups of spaces of framed embeddings of a circle in 4-manifolds.
method Investigates the fundamental groups of spaces of embeddings of S1imesD3 in 4-manifolds, focusing on framed immersed circles. result Investigates the fundamental groups of spaces of framed embeddings of a circle in 4-manifolds, providing insights into the structure of these groups.
Characterizes a class of almost Hermitian 4-manifolds using integral identities.
problem Global characterization of almost Hermitian 4-manifolds.
method Using an integral identity from Sekigawa's work, proving a uniqueness result on Lie algebras.
result Global characterization of the class AH1 of almost Hermitian 4-manifolds. 4-manifolds show every flat 3-manifold as cusp sections.
problem Realizing flat 3-manifolds as cusp sections of hyperbolic 4-manifolds.
method Transitive action on cusps, dense flat metrics realization.
result Existence of many cusp-transitive 4-manifolds.
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
Small Weyl infimum on 4-manifolds with positive scalar curvature.
problem Understanding scalar curvature on 4-manifolds.
method Analyzing the Weyl functional and comparing scalar and self-dual Weyl curvatures.
result The infimum of the Weyl functional is small on many 4-manifolds with positive scalar curvature.
The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.
problem Characterizing geometrically decomposable aspherical 4-manifolds with non-zero signature.
method Constructing examples and proving inequalities for geometrically decomposable aspherical 4-manifolds.
result All geometrically decomposable aspherical 4-manifolds with non-zero signature satisfy the inequality \( \chi \geq 3|σ| \).
Exotic submanifolds in 4-manifolds remain exotic after stabilizations.
problem Constructing exotic codimension-1 submanifolds in 4-manifolds.
method Constructing pairs of exotic codimension-1 submanifolds with diffeomorphic complements and showing they remain exotic after stabilizations.
result Exotic submanifolds remain exotic after any number of stabilizations.
Homological stability fails for 4-manifold moduli spaces, detected by new class.
problem Homological instability for moduli spaces of 4-manifolds.
method Used Seiberg-Witten equations to construct a new characteristic class.
result Homological discrepancy between smooth and topological 4-manifold moduli spaces.
Classifies Einstein metrics on 4-manifolds with specific symmetry groups.
problem Classifying Einstein metrics on 4-manifolds with certain symmetry properties.
method Analyzes cohomogeneity-one Einstein metrics and uses symmetry properties.
result Locally symmetric or homothetic to the Page metric on CP2♯CP2. 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.
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. Extends ASD connection existence to 4-manifolds with cylindrical ends.
problem Existence of ASD connections on 4-manifolds with specific ends.
method Extends gluing theorems to cylindrical ends, using mASD connections.
result Establishes existence of ASD connections on 4-manifolds with cylindrical ends.
Study exotic phenomena in small 4-manifolds using diffeomorphisms.
problem Understanding exotic smooth structures and embeddings in 4-manifolds.
method Using constraints from Seiberg-Witten theory on diffeomorphisms.
result New methods to detect and bound exotic phenomena.
New module constructs exotic surfaces in 4-manifolds.
problem Constructing exotic surfaces in 4-manifolds.
method Using Bar-Natan Khovanov homology and skein lasagna gluing maps.
result Exotic surfaces require at least one internal stabilization.
Study compact PL 4-manifolds with special handle decompositions.
problem Existence of special handlebody decompositions for simply-connected closed PL 4-manifolds.
method Investigate colored triangulations inducing handle decompositions without 1-handles or 1- and 3-handles.
result Detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary that admit such decompositions.
New flag-no-square 4-manifolds discovered with unique triangulations.
problem Identifying 4-manifolds with specific triangulation properties.
method Introduced star-connected-sum operation to preserve flag-no-square property; derived new constructions.
result Found non-aspherical flag-no-square 4-manifolds, answering a question by Przytycki and Swiatkowski.
No algorithm exists to decide 4-manifold homeomorphism.
problem Deciding homeomorphism for 4-manifolds.
method Demonstrated through a specific example of a connected sum of 12 copies of S^2 × S^2.
result No algorithm exists for deciding homeomorphism of 4-manifolds.
New results on localization of exotic diffeomorphisms in 4-manifolds.
problem Understanding when groups of exotic diffeomorphisms can be localized to smaller embedded submanifolds.
method Analyzing families Seiberg-Witten theory, Dehn twists, and properties of Seifert fibered homology spheres.
result Exotic diffeomorphisms cannot be localized to topologically embedded rational homology balls or homology spheres.
We mostly determine which closed smooth oriented 4-manifolds fibering over lower dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds.
Research examines when 4-manifolds are dominated by geometric ones.
problem When is an orientable closed 4-manifold dominated by another?
method Focuses on geometric or fibred cases.
result Characterizes conditions for domination.
Sphere bundles over 4-manifolds are trivial after looping, except for two cases.
problem Understanding the triviality of sphere bundles over 4-manifolds.
method Analyzing the splitting of sphere bundles after looping.
result The loop spaces of total manifolds of sphere bundles are homotopy equivalent, except for two special cases.
Algorithm decides homeomorphism of 4-manifolds.
problem Deciding homeomorphism of closed, simply connected 4-manifolds.
method Represent 4-manifolds by Kirby diagrams, compute Kirby-Siebenmann invariant, and use algorithms for Kirby diagrams and intersection forms.
result Algorithmic decision for homeomorphism of 4-manifolds.
New invariant for classifying 4-manifolds up to cobordism.
problem Classifying closed, oriented topological 4-manifolds up to s-cobordism. method Introducing a stable range invariant after stabilization by a fixed number of S2imesS2. result A new invariant for classifying 4-manifolds up to s-cobordism. The paper sets lower bounds for a Kirby-Thompson invariant of 4-manifolds.
problem Determining the Kirby-Thompson invariant of specific 4-manifolds.
method Using trisections, the paper establishes lower bounds and calculates the invariant for specific examples.
result The paper calculates the Kirby-Thompson invariant of the spin of L(2,1) and shows the existence of 4-manifolds with arbitrarily large invariants. In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtl…
Classifies 4-manifolds with elementary amenable groups and their boundaries.
problem Characterizing compact aspherical 4-manifolds with elementary amenable fundamental groups.
method Classification based on fundamental group properties and Farrell-Jones Conjecture.
result Such manifolds are either polycyclic or solvable Baumslag-Solitar groups.
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.
This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
problem Detecting exotic diffeomorphisms on reducible 4-manifolds with odd b+.
method Using a families Bauer-Furuta invariant and pin-cobordism.
result Constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
The study proves exotic smooth structures and equivalent genus functions in 4-manifolds.
problem Understanding exotic smooth structures and genus functions in 4-manifolds.
method Proving the existence of infinitely many exotic structures and equivalence of genus functions.
result Genus functions of exotic 4-manifolds are pairwise equivalent and stable under certain operations.
New examples of knotting phenomena in 4-manifolds with specific fundamental groups.
problem Finding 2-spheres in simply connected 4-manifolds with prescribed fundamental groups.
method Construction of infinite sets of pairwise smoothly inequivalent 2-spheres that are topologically isotopic.
result First known examples of knotting phenomena in 4-manifolds with specific properties.
Software simplifies triangulations of 4-manifolds, revealing exotic structures.
problem Understanding smooth 4-manifolds from discrete and algorithmic perspectives.
method New software tools, including an algorithm for triangulations from Kirby diagrams and a heuristic for simplification.
result Presented new triangulations of exotic pairs, corks, and plugs, including the smallest known K3 surface.
The study explores conditions for h-cobordisms between smooth 4-manifolds.
problem Determining when smoothly h-cobordant 4-manifolds are also s-cobordant. method Providing new conditions for positive answers to the s-cobordism theorem. result Conditions under which standard methods to construct h-cobordisms fail.