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 $\CP^3$.
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.
New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
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…
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.
New 4D shapes found that defy smoothness rules.
problem Smoothness rules in 4D shapes don't always hold.
method Applied reflection group trick to exotic 4-manifolds.
result Found counterexamples to smooth Borel conjecture.
We define a diffeomorphism invariant of smooth 4-manifolds which we can estimate for many smoothings of R^4 and other smooth 4-manifolds. Using this invariant we can show that uncountably many smoothings of R^4 support no Stein structure. (Gompf has constructed uncountably many smoothings of R^4 which do support Stein …
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.
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.
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.
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.
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. We show that, under a certain condition, contact 5-manifolds can `coarsely' distinguish smooth structures on compact Stein 4-manifolds via contact open books. We also give a simple sufficient condition for an infinite family of Stein 4-manifolds to have an infinite subfamily of pairwise non-diffeomorphic Stein 4-manifo…
We describe a collection of constructions which illustrate a panoply of ``exotic'' smooth 4-manifolds.
The study finds smooth structures on specific 4-manifolds with even fundamental groups.
problem Investigating smooth structures on 4-manifolds with even fundamental groups.
method Analyzing topological, closed, oriented, non-spin 4-manifolds with given constraints.
result Existence of either none or infinitely many distinct smooth structures on specified manifolds.
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 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.
Embeds 3-manifolds in symplectic 4-manifolds with constraints.
problem Embedding 3-manifolds in symplectic 4-manifolds with topological and smooth properties.
method Topological and smooth embeddings, using homology cobordism and obstructions.
result 3-manifolds can be embedded in symplectic 4-manifolds with specific conditions.
The paper generalizes a result on smooth mapping class groups and proves a property of Dehn twists in 4-manifolds.
problem Properties of Dehn twists in smooth 4-manifolds.
method Generalization of a previous result and alternative proof of a consequence.
result Dehn twists along the boundary of simply-connected 4-manifolds are trivial after connected sums.
Motivated by Stipsicz and Szabó's exotic 4-manifolds with b_2^+=3 and b_2^-=8, we construct a family of simply connected smooth 4-manifolds with b_2^+=3 and b_2^-=8. As a corollary, we conclude that the topological 4-manifold 3{CP}^2#8{-CP}^2 admits infinitely many distinct smooth structures.
We introduce a new generalization of Gompf nuclei and give applications. We construct infinitely many exotic smooth structures for a large class of compact 4-manifolds with boundary, regarding topological invariants. We prove that a large class of closed 3-manifolds (including disjoint unions of Stein fillable 3-manifo…
For a 4-manifold represented by a framed knot in S3, it has been well known that the 4-manifold admits a Stein structure if the framing is less than the maximal Thurston-Bennequin number of the knot. In this paper, we prove either the converse of this fact is false or there exists a compact contractible oriented smo…
Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.
problem Proving topological and smooth pseudo-isotopies of simply connected 4-manifolds.
method Provided different arguments that bypass the replacement criterion, thus completing Quinn's proofs.
result Corrected and completed Quinn's proofs of both topological and stable smooth pseudo-isotopy theorems.
The paper creates exotic 4-manifold structures with a specific group.
problem Producing exotic structures on 4-manifolds with infinite dihedral fundamental group.
method Using specific conditions on b2+ and b2−, the paper constructs these structures. result The existence of infinite exotic structures on 4-manifolds with infinite dihedral fundamental group.
New definite 4-manifolds found with non-cyclic groups.
problem Finding exotic smooth structures on 4-manifolds with specific fundamental groups.
method Constructing infinitely many non-diffeomorphic structures.
result Infinitely many pairwise non-diffeomorphic definite 4-manifolds with Z/2imesZ/2 fundamental group. Paper studies smoothness of bi-conformal heat flow on 4-manifolds.
problem Smoothness of bi-conformal heat flow on 4-manifolds.
method Introduces bi-conformal heat flow (bi-CHF) and proves global smoothness without finite time singularities.
result Global smoothness and no finite time singularity for bi-conformal heat flow.
The broken genera are orientation preserving diffeomorphism invariants of closed oriented 4-manifolds, defined via broken Lefschetz fibrations. We study the properties of the broken genera invariants, and calculate them for various 4-manifolds, while showing that the invariants are sensitive to exotic smooth structures…
We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.
Study on diffeotopy groups of non-compact 4-manifolds, extending previous results.
problem Understanding diffeotopy groups of non-compact 4-manifolds.
method Analyzing smoothings of punctured 4-manifolds and proving properties of diffeotopy groups.
result Uncountable diffeotopy groups for certain non-compact 4-manifolds.
We study exotic smoothings of open 4-manifolds using the minimal genus function and its analog for end homology. While traditional techniques in open 4-manifold smoothing theory give no control of minimal genera, we make progress by using the adjunction inequality for Stein surfaces. Smoothings can be constructed with …
This paper investigates the question of which smooth compact 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called OPTIMAL; Einstein metrics and scalar-flat anti-self-dual metrics provide us with two interesting classes of examples. Using twistor m…
Study on 4-manifolds with exotic smooth structures and Z_2 fundamental group.
problem Exploring smooth structures on 4-manifolds with specific fundamental groups.
method Construction of irreducible, smooth, oriented, closed, definite 4-manifolds with Z_2 fundamental group and specific Betti numbers.
result Proves existence of infinitely many smooth structures on definite 4-manifolds with positive second Betti number and Z_2 fundamental group.
Classifies 4-manifolds with specific properties and fundamental group.
problem Classifying 4-manifolds with fundamental group Z and boundary. method Topological and smooth classification methods, including Hermitian forms and 2-handlebody constructions. result Every Hermitian form over Z[t±1] arises as the equivariant intersection form of exotic smooth 4-manifolds. Simply-connected 4-manifolds are covered by products with a 2-torus.
problem Proving that any closed simply-connected smooth 4-manifold is branched covered by a product of an orientable surface and a 2-torus.
method Natural construction with respect to spin structures, solving Problem 4.113(C) in Kirby's list.
result Closed simply-connected smooth 4-manifolds are 16-fold branched covered by a product of an orientable surface and a 2-torus.
It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard R4. Similarly, a smooth 4-manifold homeomorphic to the produc…
We construct noncomplex smooth 4-manifolds which admit genus-2 Lefschetz fibrations over S^2. The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not nece…
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.
Smooth surfaces in simply connected 4-manifolds yield groups with non-trivial homology.
problem Finding groups with non-trivial fundamental groups in the complements of surfaces in 4-manifolds.
method Combining topological constructions with homological properties of simple groups.
result The complement of a surface in a simply connected 4-manifold can have a non-trivial fundamental group.
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.
The study examines the realizability of a 4-manifold invariant for homeomorphisms.
problem Realizing the Casson-Sullivan invariant for homeomorphisms of 4-manifolds.
method Investigation of the invariant's realizability and application to surface isotopy.
result The invariant can be realized fully after stabilizing with a single S2imesS2 for all orientable pairs of homeomorphic 4-manifolds. Paper examines 4-manifold structures using chord assignments.
problem Determining if knot surgery 4-manifolds are diffeomorphic.
method Assigns integers to chords in crown diagrams to compare smooth structures.
result Smooth structures on knot surgery 4-manifolds are not isotopic.
Study constraints on diffeomorphisms and homeomorphisms of 4-manifolds with boundary.
problem Constraints on smooth families of 4-manifolds with boundary.
method Use Manolescu's Seiberg-Witten Floer stable homotopy type.
result Inclusion map between diffeomorphisms and homeomorphisms is not a weak homotopy equivalence.
For suitable finite groups G, we construct contractible 4-manifolds C with an effective G-action on ∂C whose associated pairs (C,g) for all g∈G are distinct smoothings of the pair (C,∂C). Indeed C embeds in a 4-manifold so that cutting out C and regluing using distinct elements of G yield dist…
Smoothly isotopic surfaces in a 4-manifold are stable isotopic if trivial in the 2-homology.
problem Investigating whether topologically isotopic surfaces are smoothly isotopic in stabilised 4-manifolds.
method Analyzing the triviality of surfaces in the 2-homology of the ambient 4-manifold and constructing stabilisations.
result The result holds for surfaces trivial in the 2-homology of the 4-manifold and for certain fundamental groups.
Two types of 4-manifolds disagree on minimum Euler characteristic.
problem Disagreement between topological and smooth 4-manifolds.
method Comparison of topological and smooth 4-manifolds with a fixed fundamental group.
result Minimum Euler characteristic differs between topological and smooth 4-manifolds.
New smooth structures found on certain 4D spaces.
problem Finding distinct smooth structures on specific 4D spaces.
method Constructing specific 4-manifolds with infinite fundamental group.
result Infinitely many non-diffeomorphic structures found.
In this paper we prove several related results concerning smooth Zp or $\s^1$ actions on 4-manifolds. We show that there exists an infinite sequence of smooth 4-manifolds Xn, n≥2, which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each Xn support…
A proof via the Seiberg-Witten moduli space of Donaldson's theorem on smooth 4-manifolds with definite intersection forms.