Study knotted surfaces in simply-connected 4-manifolds with trivial boundary.
problem Locally flat, compact, genus g surfaces with boundary a fixed knot in simply-connected 4-manifolds with boundary the 3-sphere.
method Analyzes homology and isotopy of surfaces with trivial fundamental group complements.
result Homologous surfaces are topologically ambiently isotopic rel. boundary.
New homotopy 4-spheres and real projective 4-spaces created.
problem Creating new homotopy 4-spheres and real projective 4-spaces.
method Extending Cappell-Shaneson's construction to produce new sets of smooth 4-manifolds.
result Produces new collections of homotopy 4-spheres and real projective 4-spaces.
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 paper tackles deeply slice knots via immersed curves.
problem Whether there exists a homology 3-sphere with certain knot properties.
method Constructing knots in contractible 4-manifolds and using Heegaard Floer homology invariants.
result Proves the existence of a class of deeply slice knots.
The paper calculates a new invariant for 4-manifolds using handle decompositions and skein relations.
problem Computing invariants for 4-manifolds built from handles.
method Handle attachment formulas, cabled colimits, lasso relation.
result Explicit calculations and partial vanishing results for specific 4-manifolds.
Dehn twists on K3-type 4-manifolds are not homotopy coherently Nielsen realizable.
problem Homotopy coherent Nielsen realization problem for Dehn twists on 4-manifolds
method Using family Seiberg-Witten theory
result Failure of the classical Nielsen realization problem in K3-type 4-manifolds
Construct non-split 2-component links in S4 to produce exotic pairs of 4-manifolds
problem Construct non-split 2-component links in S4 method Construct non-split 2-component links in S4 result Obtain exotic pairs of simply connected, definite 4-manifolds with boundary
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
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.
New method uses Floer homology to create exotic disk pairs.
problem Creating exotic disk pairs in 4-manifolds.
method Singular instanton Floer homology with Chern--Simons filtration.
result Constructs exotic pairs of slice disks and knots.
For a specific class of 4-manifolds, random isometries cannot lift to orientation-preserving diffeomorphisms.
problem When does a finite group of isometries of a specific 4-manifold lift to orientation-preserving diffeomorphisms?
method Combination of equivariant connected-sum constructions, fixed-point theory, finite group actions on surfaces, analytic combinatorics, and previous work.
result Random subgroups of isometries are asymptotically almost never realizable in orientation-preserving diffeomorphisms.
Simple branched coverings established between certain 4-manifolds.
problem Establishing simple branched coverings between specific 4-manifolds.
method Using handle decompositions and isometric embeddings of intersection lattices.
result Existence and construction of d-fold simple branched coverings. New singularities and fibrations in non-orientable 4-manifolds.
problem Understanding singularities and fibrations in non-orientable 4-manifolds.
method Introducing M-singularities and M-fibrations, studying their handle decompositions and orientation double coverings. result Relations among crosscap transpositions give rise to M-fibrations on non-orientable 4-manifolds. New method for quandle presentations of surface knots in 4-manifolds.
problem Computing fundamental quandle presentations for surface knots in arbitrary 4-manifolds.
method Wirtinger type presentation of the fundamental quandle for surface links in 4-manifolds.
result Infinitely many pairwise non-local surface knots with specific bridge numbers in certain 4-manifolds.
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. Surface corks modify 4-manifold structures without changing their homeomorphism type.
problem Understanding how to change 4-manifold structures using surface corks.
method Introducing surface corks as compact, contractible submanifolds intersecting smoothly embedded surfaces in 4-manifolds.
result Explicit construction of a transverse surface cork that is diffeomorphic to a 4-ball.
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.
New invariants for 4-manifolds obstruct certain surface pairs.
problem Obstructing the existence of specific surface pairs in 4-manifolds.
method Defining new mixed invariants using involutive Heegaard Floer homology.
result The new invariants obstruct the existence of certain surface pairs.
Survey of gauge theory for families of 4-manifolds.
problem Understanding gauge theory for families of 4-manifolds.
method Developed gauge theory for diffeomorphism groups of 4-manifolds.
result New applications of gauge theory to diffeomorphism groups.
Gauge theory for families helps compare 4-manifold groups.
problem Comparing diffeomorphism and homeomorphism groups of 4-manifolds.
method Gauge theory applied to families of 4-manifolds.
result Improved understanding of group comparisons in 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.
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 proof shows exotic 4-manifolds exist without complex calculations.
problem Existence of exotic 4-manifolds in topology.
method Used Beliakova and Wehrli's s-invariant for links in S3 and Stošić's induction scheme to simplify computations. result Existence of exotic compact, orientable 4-manifolds proven without skein lasagna modules.
Formulae for 1-3 handle attachments in 4-manifolds.
problem Understanding handle attachments in 4-dimensional manifolds.
method Developed general formulae for 1-3 handle attachments using skein modules.
result Derived complete description of gluing homomorphism on skein modules.
New invariant for 4-manifolds with framed links, stronger than existing invariants.
problem Distinguishing 4-manifolds with framed links.
method Introducing a new invariant called KLS lasagna homotopy type.
result The new invariant is stronger than existing invariants.
The paper extends knot theory to 4-manifolds, defining new genera and obstructions.
problem Understanding embeddings of 3-manifolds into 4-manifolds with specific properties.
method New L2-signature obstructions and extensions of knot genera. result Existence of knots with arbitrarily large generalized superslice genera and double stabilizing numbers.
The paper classifies surfaces in 4-manifolds up to concordance.
problem Classifying surfaces in 4-manifolds up to concordance.
method Ambient surgery with Pin−-structures, cobordism theory, and topological constructions. result Closed surfaces in simply-connected 4-manifolds are concordant if cobordant.
New 4-manifolds without 1- and 3-handles are created from knots.
problem Creating 4-manifolds without specific handles. method Knot surgery with specific knots.
result Knot surgery 4-manifolds E(n)K have a handle decomposition without 1- and 3-handles. Geometrically, a new obstruction is found for 4-manifold realizations.
problem Realizing normal 1-types of 4-manifolds with a given boundary.
method Three-stage obstruction theory, with a geometric tertiary obstruction.
result A new geometric tertiary obstruction for 4-manifold realizations.
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 classification of nonorientable 4-manifolds with specific fundamental groups.
problem Classifying nonorientable 4-manifolds with cyclic fundamental groups.
method Simple cut-and-paste construction, using results from Hambleton-Kreck-Teichner and Khan.
result Plausible classification of a large set of nonorientable 4-manifolds with cyclic fundamental groups of order 2p.
The paper extends torus surgery results in 4-manifolds and shows diffeomorphisms.
problem Extending torus surgery results in 4-manifolds.
method Analyzing 4-manifolds diffeomorphic to S1 times exterior of fibered knots and T2imesD2 bundles over T2. result The result of torus surgery in S1imesY along S1imesK fibers over S1. New invariants detect exotic smooth structures in 4-manifolds.
problem Detect exotic smooth structures in 4-manifolds using invariants of 2-handlebodies.
method Investigates invariants of 4-dimensional 2-handlebodies from the Temperley-Lieb category in positive characteristic.
result Height n=2 invariant vanishes on CP2, CP2, and S2imesS2 for p>3. Uniform proof of π1-injectivity for certain maps in low dimensions.
problem Characterizing π1-injective maps in low-dimensional manifolds. method Uniform group-theoretic proof based on residual finiteness and numerical invariants.
result Uniform determination of when maps induce π1-isomorphisms. 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.
The paper establishes criteria for symplectic surfaces in 4-manifolds.
problem Finding symplectic surfaces in 4-manifolds with bounded genus.
method Criterion for minimization of genus among surfaces with given homology and boundary.
result Symplectic surfaces minimize slice genus for their boundary in contact 3-manifolds.
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.
The paper studies exotic diffeomorphisms of 4-manifolds with b_+ = 2.
problem Examining exotic diffeomorphisms in 4-manifolds with a specific topological invariant.
method Utilizes Seiberg-Witten invariants for 1-parameter families of 4-manifolds and a gluing formula for connected sums.
result Proves that the mapping class group of certain 4-manifolds is not finitely generated and surjects to Z^∞.
E-string theory reveals modular properties of 4-manifold invariants.
problem Understanding the topology of 4-dimensional manifolds.
method Computing partition function on M4imesT2 and verifying its modular properties. result The partition function of the E-string theory is modular and can be lifted to a topological modular form.
This thesis constructs families of arcs in 4-manifolds and analyzes their homotopy properties.
problem Analyzing the homotopy properties of families of embedded arcs in 4-manifolds.
method Using embedding calculus and diagrammatic frameworks inspired by cubical ω-groupoids, the thesis constructs and analyzes families of embedded arcs.
result The family G(p,q,r) is trivial in π3T3Emb∂(I,M) but conjectured to be non-trivial in π3T4Emb∂(I,M). The paper classifies 4-manifolds with given boundaries.
problem Extending homotopy equivalences of boundaries to 4-manifolds.
method Surgery theory and classification of groups.
result Conditions for extending homotopy equivalences to homeomorphisms.
Defines a new invariant for 4-manifolds with boundary.
problem Homotopy classification of 4-manifolds with boundary.
method Definition of a relative k-invariant for CW pairs.
result Relative k-invariant is an obstruction to extending maps.
Researchers construct irreducible 4-manifolds with specific properties.
problem Creating smooth manifolds with specific topological and geometric properties.
method Equivariant fiber sums of Lefschetz fibrations and symplectic manifolds.
result Constructed irreducible manifolds with even intersection forms and specific topological invariants.
Proves a formula for a special invariant of 4-manifolds.
problem Calculating the Bauer-Furuta invariant for connected sums of 4-manifolds.
method Uses a finite dimensional approximation of the Seiberg-Witten monopole map to derive a formula for the families Bauer-Furuta invariant of a fibrewise connected sum.
result Derives a general connected sum formula for the families Bauer-Furuta invariant.
New method uses Khovanov homology to distinguish exotic 4-manifolds.
problem Distinguishing exotic compact, orientable 4-manifolds.
method Introduces a new, simple way of using Khovanov homology.
result First analysis-free proof of existence of exotic Mazur 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.
New modules derived from Khovanov homology for links.
problem Constructing new link homology modules from Khovanov homology.
method Functoriality proof for cobordisms in 4D relative 1-handlebody complements.
result Functoriality of Rozansky-Willis's homology for cobordisms.
The paper studies mapping class groups of nontrivial S2 fiber bundles.
problem Analyzing the mapping class groups of nontrivial S2 fiber bundles. method Using generalizations of Dax invariants for embedded surfaces in 4-manifolds.
result Surjective homomorphisms from MCG(X) and MCG(X′) to Z∞ are shown. New 4D shapes can't be opened like books.
problem Existence of 4D shapes without open book decompositions.
method Demonstrated existence of infinitely many parallelizable 4-manifolds without open book decompositions.
result No open book decomposition for certain Engel manifolds.
New mathematical proposal for TQFTs using TMF-modules.
problem Constructing new types of TQFTs at the intersection of topology, algebra, physics, and homotopy theory.
method Defines TMF-modules associated with symmetric bilinear forms and assigns them to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms.
result Invariants of 4-manifolds arising from 6-dimensional superconformal field theories, conjecturally generalizing the theta function of a lattice.
Non-orientable 4-manifolds are simple branched coverings of RP^4 and twisted S^3-bundles.
problem Characterizing non-orientable 4-manifolds as branched coverings.
method Showing that every closed connected non-orientable PL 4-manifold is a simple branched covering of $\RP^4$ and a twisted S3-bundle. result Non-orientable 4-manifolds are simple branched coverings of $\RP^4$ and twisted S3-bundles, with specific conditions on the degree and branch set. Link homology theories connect to 4-manifold invariants and TQFTs.
problem Connecting link homology theories to 4-manifold invariants and TQFTs.
method Functorial link homologies, skein modules, handle decompositions.
result Link homology theories become diagram-independent and furnish invariants of 4-manifolds.
A new method studies symplectic configurations in rational 4-manifolds using computer-aided techniques.
problem Understanding symplectic configurations in rational 4-manifolds.
method Computer-aided approach combining Cremona transformations and pseudoholomorphic curves.
result Nonexistence of Fano planes in the symplectic category.
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.
Study shows how any group can be fundamental of a non-orientable 4-manifold with exotic structure.
problem Realizing any finitely presented group as the fundamental group of a non-orientable 4-manifold with exotic structure.
method Performing a Gluck twist to obtain an exotic smooth structure, and using 2-covers to stabilize the structure.
result Any finitely presented group can be realized as the fundamental group of a non-orientable 4-manifold with exotic structure.
New stable exotic 4-manifolds found with specific topological properties.
problem Identifying conditions for the existence of stable exotic 4-manifolds.
method Investigated fundamental group, Stiefel-Whitney classes, and H_5(π;Z) to determine stable exotic pairs.
result Produced new stable exotica and settings where they do not arise.
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.
New findings show infinitely many knots cannot be smoothly round handle slices.
problem Understanding the smoothability of knots in 4-dimensional space.
method Analyzing the properties of knots under surgery conjectures and cobordism conjectures.
result Infinitely many knots fail to be smoothly round handle slices.
Study on topological pseudo-isotopies of 4-manifolds, proving some cases and finding counterexamples.
problem Determining when a topologically pseudo-isotopic diffeomorphism is also smoothly pseudo-isotopic.
method Analyzing the fundamental group of 4-manifolds and constructing counterexamples.
result Some fundamental groups guarantee a positive answer, while others can have negative outcomes.