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. 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…
We prove that for each closed smooth spin 4-manifold M there exists a closed smooth 4-manifold N such that the connected sum M # N admits a conformally flat Riemannian metric.
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.
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.
Closed Lorentz 4-manifolds have finite isometry groups with a bounded abelian subgroup.
problem Understanding the structure of isometry groups of closed Lorentz 4-manifolds.
method Proving that any finite subgroup of the isometry group of a closed Lorentz 4-manifold has a bounded abelian subgroup.
result Finite isometry groups of closed Lorentz 4-manifolds have a bounded abelian subgroup.
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.
Develops technique to trisect 4-manifolds using Lefschetz fibrations.
problem Trisecting 4-manifolds with nonempty boundary.
method Gluing relative trisection diagrams of 4-manifolds with Lefschetz fibrations.
result Explicit trisection diagrams for various 4-manifolds.
Paper finds exotic diffeomorphisms on specific 4-manifolds.
problem Understanding exotic diffeomorphisms on 4-manifolds with b2+=2. method Comparing winding numbers of parameter families.
result 2\mathbb{C}\mathbb{P}^2 \# 10 (-\mathbb{C}\mathbb{P}^2) admits exotic diffeomorphisms.
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.
We prove that for any \e>0, there exists a closed hyperbolic 4-manifold with a closed geodesic of length < \e.
Paper finds first hyperbolic 4-manifold with harmonic spinors.
problem Finding harmonic spinors on hyperbolic 4-manifolds.
method Using G-spin theorem and explicit calculations for spin hyperbolic manifolds.
result First example of a closed hyperbolic 4-manifold with harmonic spinors.
Closed 4-manifolds foliated by hyperplanes are homeomorphic to the 4-torus.
problem Closed 4-manifolds foliated by hyperplanes
method Prove that such manifolds are homeomorphic to the 4-torus
result Closed 4-manifolds foliated by hyperplanes are homeomorphic to the 4-torus
Constructs 4-manifolds with positive Euler characteristic proving a conjecture.
problem Finding aspherical 4-manifolds with positive Euler characteristic.
method Explicit construction of 4-manifolds with specified properties.
result Proves a conjecture by constructing a 4-manifold with Euler characteristic 1.
A small projective 4-manifold created via Dehn filling.
problem Creating a small positive Euler characteristic closed convex projective 4-manifold.
method Explicit construction through continuous path of projective cone-manifolds and Dehn filling of a cusped hyperbolic 4-manifold.
result Obtained a closed orientable convex projective four-manifold with small positive Euler characteristic.
Plumbing of surfaces embeds in hyperbolic 4-manifolds.
problem Embedding specific geometric structures in hyperbolic manifolds.
method Using plumbings of disc bundles over surfaces.
result Plumbed structures can be embedded in closed hyperbolic 4-manifolds.
In this paper we describe the classification of all the geometric fibrations of a closed flat Riemannian 4-manifold over a 1-orbifold.
This is a survey paper on the space of symplectic structures on closed 4-manifolds, for the Proceedings ICCM 2004
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.
New examples of 4-manifolds distinguished by knot slicing.
problem Distinguishing closed 4-manifolds using knot slicing.
method Finding knots that are smoothly slice in one manifold but not the other.
result First examples of 4-manifolds distinguished by this approach.
Proves properties of 4-manifolds with scalar curvature constraints.
problem Characterizing 4-manifolds with specific scalar curvature properties.
method Analyzes locally conformally flat metrics and uses Schoen's conjecture.
result Affirmatively answers Noronha's question about 4-manifolds with scalar curvature zero.
Extends 4D cornered skein theory to surfaces, proving gluing formulas.
problem Formulating gluing formulas for 4-manifolds with corners and boundaries.
method Develops a categorical framework and introduces bicategories for closed surfaces.
result Proves gluing formulas for categories associated with 3-manifolds with boundary.
Smooth 4-manifolds have simple horizontal decompositions.
problem Classifying smooth, closed, orientable 4-manifolds.
method Horizontal handlebody decomposition.
result Simplest horizontal decompositions classify closed 4-manifolds.
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.
In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: S4, CP2, S2×S2, or CP2#±CP2. As an…
The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.
problem Characterizing and classifying 4-manifolds using Artin presentations and triangle groups.
method Utilizing triangle groups to find Artin presentations that present the trivial group and determining 4-manifolds with specific properties.
result Identified all Artin presentations on two generators that present the trivial group and all smooth, closed, simply-connected 4-manifolds with specific properties.
Study shows non-negative curvature on 4-manifolds with torus symmetry.
problem Classifying 4-manifolds with torus symmetry under non-negative curvature.
method Investigated invariant metrics on 4-manifolds with circle and torus actions.
result Found that almost non-negative curvature implies non-negative curvature for 4-manifolds with torus symmetry.
In this paper we determine the integral homology and cohomology groups of a closed 4-manifold X obtained as the generalized fibre sum of two closed 4-manifolds M and N along embedded surfaces of genus g and self-intersection zero. If the homologies of the 4-manifolds are torsion free and the surfaces represent indivisi…
Every 4-manifold can be smoothly embedded in complex projective 3-space.
problem Embedding 4-manifolds in complex projective spaces.
method Analyzing properties of 4-manifolds and complex projective spaces.
result Every closed orientable smooth 4-manifold admits a smooth embedding in $\CP^3$.
It is well known that for any exotic pair of simply connected closed oriented 4-manifolds, one is obtained from the other by twisting a compact contractible submanifold via an involution on the boundary. By contrast, here we show that for each positive integer n, there exists a simply connected closed oriented 4-mani…
The paper classifies 4-manifolds based on their fundamental groups and orientation characters.
problem Classifying 4-manifolds with specific fundamental groups and orientation characters.
method Developed a criterion for groups and homomorphisms to classify 4-manifolds up to homotopy equivalence.
result Closed 4-manifolds with specified fundamental groups and orientation characters are classified by their quadratic 2-types.
Closed oriented 4-manifolds with the same geometrically 2-dimensional fundamental group (satisfying certain properties) are classified up to s-cobordism by their w2-type, equivariant intersection form and the Kirby-Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed…
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.
New classification for certain 4-manifolds using quasiregular mappings.
problem Classifying closed simply connected quasiregularly elliptic 4-manifolds.
method Using quasiregular mappings and de Rham cohomology.
result Closed simply connected quasiregularly elliptic 4-manifolds are homeomorphic.
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.
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.
Paper computes G-index for specific hyperbolic manifolds.
problem Computing G-index for closed, spin, hyperbolic manifolds.
method Developed techniques to compute G-index for 2- or 4-manifolds.
result Computed G-index for Davis hyperbolic 4-manifold.
The paper shows Thurston geometries don't support Anosov diffeomorphisms.
problem The existence of Anosov diffeomorphisms on Thurston geometries.
method Analyzing Thurston geometries and their properties.
result Thurston geometries do not support transitive Anosov diffeomorphisms.
The paper smooths out equations on 4-manifolds to show smooth moduli spaces.
problem Understanding the structure of solutions to Vafa-Witten equations on 4-manifolds.
method Constructing perturbation terms to ensure transversality and showing moduli spaces are smooth.
result For generic perturbation, the general part of the moduli space is a smooth 0-dimensional manifold.
We study the quasi-convergence equivalence of some families of metrics on locally homogeneous closed 4-manifolds with trivial isotropy group, and identify the dimension of each equivalence class under certain conditions.
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.
Study on 4-manifolds restricts negatively curved metrics and fundamental groups.
problem Understanding negatively curved metrics on 4-manifolds.
method Analysis of transnormal and isoparametric functions.
result Closed 4-manifolds cannot be negatively curved.
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.
We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w_2-type and their equivariant intersection forms are stably isometric. We also find explicit algebraic invaria…
New findings on 4-manifolds without symplectic structures.
problem Identifying 4-manifolds without symplectic structures.
method Analyzing stable cohomotopy Seiberg-Witten invariants and geometric properties.
result Positive definite 4-manifolds without 1-handles have vanishing stable cohomotopy Seiberg-Witten invariants.
This paper computes trisection genus for PL 4-manifolds using triangulations.
problem Computing the trisection genus for PL 4-manifolds.
method Algorithm implementation and triangulation-based approach.
result Lower bounds on trisection genus and construction of minimal genus trisections.
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.
The paper shows maps from 3-manifolds to 4-manifolds can't preserve fundamental groups.
problem Maps from 3-manifolds to 4-manifolds and their fundamental groups.
method Analyzing properties of 3-manifolds and 4-manifolds, proving properties of maps.
result Maps from 3-manifolds to 4-manifolds do not induce isomorphisms on fundamental groups.