The study proves convergence of conic 4-spheres' geometry to boundary cases.
problem Convergence of conic 4-spheres' geometry to boundary cases.
method Proved a convergence theorem on the moduli space of constant σ₂ metrics for conic 4-spheres.
result When a numerical condition converges to the boundary case, conic 4-spheres' geometry converges to the boundary case while preserving capacity.
We discuss the constant σ2 problem for conic 4-spheres. Based on earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a necessary condition for the existence problem. In particular, when the condition is sharp, we have the uniqueness result similar to that of Troyanov in dimension 2. It indicat…
Constructs new coassociative fibrations for G2 manifolds.
problem Tackles the construction of new coassociative fibrations for G2 manifolds.
method Constructs fibrations by coassociative 4-folds, relates to hypersymplectic geometry and Donaldson's work.
result Shows natural generalizations of known coassociative fibrations.
Researchers found a family of Sp(2)-invariant solitons for Laplacian flow.
problem Analyzing Sp(2)-invariant solitons for Laplacian flow on the 4-sphere.
method Used mathematical analysis and asymptotic cone determination.
result Identified a 1-parameter family of Sp(2)-invariant expanding solitons with specific asymptotic behavior.
Every smooth 4-sphere is the same as the standard one.
problem Identifying smooth 4-spheres.
method Proved diffeomorphism to the standard 4-sphere.
result Smooth homotopy 4-spheres are diffeomorphic to the 4-sphere.
Coassociative submanifolds are 4-dimensional calibrated submanifolds in G2-manifolds. In this paper, we construct explicit examples of coassociative submanifolds in Λ−2S4, which is the complete G2-manifold constructed by Bryant and Salamon. Classifying the Lie groups which have 3- or 4-dimensional…
The paper constructs homotopy 4-spheres using pochette surgery.
problem Creating homotopy 4-spheres from pochette surgeries.
method Pochette surgery generalizes Gluck surgery to construct embeddings of pochettes into the 4-sphere and proves homotopy 4-spheres are diffeomorphic to the 4-sphere.
result Homotopy 4-spheres obtained from pochette surgeries are all diffeomorphic to the 4-sphere.
Paper proves every stable 4-sphere has a unique diffeomorphism class.
problem Identifying stable 4-spheres and their diffeomorphisms.
method Using Wall's result and properties of surface-knot spaces.
result Every stable 4-sphere has a unique orientation-preserving diffeomorphism class.
Calegari's 4-spheres from fibered knots are proven standard.
problem Proving Calegari's 4-spheres from fibered knots are standard.
method 5-dimensional handlebody techniques and mapping class groups of 3-dimensional handlebodies.
result All Calegari's homotopy 4-spheres from fibered knots are diffeomorphic to the standard 4-sphere.
Study pochette surgery on 4-manifolds, focusing on 4-spheres.
problem Understanding pochette surgery on 4-spheres and its effects.
method Using linking number of pochette embeddings, compute homology and analyze surgeries.
result Pochette surgery on any homology 4-sphere can be computed via homology, and trivial cord surgeries do not change diffeomorphism type.
The Price twist creates three 4-manifolds from a 4-sphere.
problem Understanding the properties of a non-simply connected 4-manifold created from a 4-sphere.
method Cutting and pasting operation on a P2-knot S in a 4-manifold. result The non-simply connected 4-manifold τS(S4) is studied for Kinoshita type P2-knots. A surface in the 4-sphere is trivially embedded, if it bounds a 3-dimensional handle body in the 4-sphere. For a surface trivially embedded in the 4-sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.
Study on Ricci flow on 4-spheres, proving standard sphere convergence.
problem Characterizing and understanding Ricci flow on 4-spheres.
method Investigation of integral conformal invariants, analysis of flow properties.
result Established monotonic decay of certain curvature norms, leading to standard sphere convergence.
Turing complete flow on 4-sphere preserves volume.
problem Creating a Turing complete flow on a 4-sphere.
method Smooth, conservative flow on the 4-sphere.
result Achieved a Turing complete, volume-preserving flow.
New theory proves infinite homology 3-spheres in homology 4-spheres.
problem Existence of homology 3-spheres in homology 4-spheres.
method Diagrammatics of surface cross sections, Taubes' work.
result Infinite number of homology 3-spheres in homology 4-spheres.
Researchers found non-smoothable surfaces in a 4-sphere, solving K3 problems.
problem Non-smoothable surfaces in the 4-sphere.
method Constructed non-orientable surfaces with specific knot groups.
result Found surfaces that are non-smoothable and answered K3 problems.
Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between…
Gluck twisting certain knots results in standard 4-spheres.
problem Understanding when Gluck twists yield standard 4-spheres.
method Analyzing smooth homotopy 4-spheres and diffeomorphisms.
result Infinite collection of twisted doubles of corks are standard.
New spanning 3-disks found for unlink in 4-sphere.
problem 2-component unlink in 4-sphere.
method Found infinitely many isotopy classes of Brunnian spanning 3-disks.
result Infinitely many Brunnian spanning 3-disks for 2-unlink in 4-sphere.
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.
Smoothly knotted 5RP^2 found in 4-sphere.
problem Finding knotted embeddings in higher dimensions.
method Topological and smooth knotting analysis in 4-sphere.
result Smoothly knotted 5RP^2 found in 4-sphere.
Examples are given to show that some compact contractible 4-manifolds can be knotted in the 4-sphere. It is then proved that any finitely presented perfect group with a balanced presentation is a knot group for an embedding of some contractible 4-manifold in the 4-sphere.
Standardizes Dunfield-Gong's 4-sphere, solves knot sliceness problem.
problem Standardizing Dunfield-Gong's 4-sphere and solving knot sliceness problem.
method Standardization and fibered handle-ribbon disk construction.
result 18_{nh00000601} knot is slice in the standard 4-ball.
Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
problem Smooth isotopy of 3-discs in 4-sphere.
method Pushing 3-discs into 5-dimensional space.
result Isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
We prove that the infinite family of homotopy 4-spheres constructed by Daniel Nash are all diffeomorphic to 4-sphere.
New research finds 145 infinite families of CS spheres are standard.
problem Determining which Cappell-Shaneson spheres are diffeomorphic to the standard 4-sphere.
method Using Kirby calculus and new families of CS spheres.
result Proves 145 new infinite families of CS spheres are standard.
New findings about twists in 4-sphere diffeomorphisms.
problem Understanding isotopy classes of diffeomorphisms in 4-sphere.
method Using Cerf theory and twists along Montesinos twins.
result The subgroup of twists along Montesinos twins is trivial or cyclic of order two.
3-balls in 4-sphere become isotopic in 5-ball.
problem Whether 3-balls in 4-sphere become isotopic in 5-ball.
method Analyzing the embedding of 3-balls in 4-sphere and 5-ball.
result Affirmative answer to Gay, Hughes, Kim, and Miller's question.
We present an infinite sequence of smooth embeddings of a connected sum of 6 projective planes in the 4-sphere, which are all ambient homeomorphic, but pairwise ambient non-diffeomorphic. The double covers of the 4-sphere ramified along these surfaces form a family of the exotic $\Bbb CP^2#5\bar{\Bbb CP^2}$ constructed…
Generators of the 4-sphere's smooth mapping class group via diffeomorphisms of Montesinos twins.
problem Understanding the smooth mapping class group of the 4-sphere.
method Homomorphism from loops of 2-spheres to smooth mapping class group, generators as twists along Montesinos twins.
result Generators of the image of the homomorphism as diffeomorphisms of Montesinos twins.
In this paper, Problem 4.17 on R. Kirby's problem list is solved by constructing infinitely many aspherical 4-manifolds that are homology 4-spheres
Researchers create infinite Brunnian links of 3-balls in 4-sphere.
problem Constructing infinite Brunnian links of 3-balls in 4-sphere.
method Using the third author's result on splitting spheres for trivial 2-spheres link in 4-sphere, and providing a new proof.
result Infinitely many n-component Brunnian links of 3-balls in 4-sphere constructed.
Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.
problem Understanding the structure of 2-complexes and their asphericity.
method Using Kervaire's sphere-link and ribbon sphere-link equivalence, analyzing the compact complement of ribbon disk-links.
result Every connected subcomplex of a contractible finite 2-complex is aspherical.
In relation to the 4-dimensional smooth Poincaré conjecture we construct a tentative invariant of homotopy 4-spheres using embedded contact homology (ECH) and Seiberg-Witten theory (SWF). But for good reason it is a constant value independent of the sphere, so this null-result demonstrates that one should not try to us…
Infinite Klein bottles with 4-fold meridians found.
problem Finding indecomposable Klein bottles with specific meridian orders.
method Exhibited an infinite family of Klein bottles in 4-sphere.
result Infinite family of indecomposable Klein bottles with order-4 meridians.
The only finite nonabelian simple group acting on a homology 3-sphere - necessarily non-freely - is the dodecahedral group A5≅PSL(2,5) (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group A5∗≅SL(2,5)). In the present pa…
This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued function theory, whose "meromorphic functions" are conformal maps into quaternions, wh…
We characterize Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic curves in complex projective space or as inverted minimal tori (with planar ends) in Euclidean 4-space.
Explains a 1978 construction for Yang-Mills instantons.
problem None explicitly stated; focuses on explaining a construction.
method Atiyah-Drinfeld-Hitchin-Manin construction
result Explains the ADHM construction of Yang-Mills instantons.
For a non-orientable closed surface standardly embedded in the 4-sphere, a diffeomorphism over this surface is extendable if and only if this diffeomorphism preserves the Guillou-Marin quadratic form of this embedded surface.
We show that a finite group which admits a faithful, smooth, orientation-preserving action on a homology 4-sphere, and in particular on the 4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5), by explicitly determining the various groups which can occur (up to an indetermination of index two in the case…
New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.
problem Existence of non-isotopic handlebodies with identical boundary.
method Construction of specific genus-g handlebodies in S4 and B5. result Proves Budney-Gabai conjecture for genus at least 2.
We prove that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere is either of "finite type", that is, has a spectral curve of finite genus, or is of "holomorphic type" which means that it is super conformal or Euclidean minimal with planar ends. This implies that all constrained Willmore tori in …
For every N > 0 there exists a group of deficiency less than -N that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold o…
This is a collection of notes on embedding problems for 3-manifolds. The main question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 mani…
Study G2-instantons on 7-sphere from 4-sphere instantons.
problem Deformation theory of G2-instantons on 7-sphere. method Analysis of pullback from 4-sphere instantons and identification of deformation spaces.
result Existence of a smooth, complete family of 15-dimensional G2-instantons. The paper constructs many knotted and linked objects in higher dimensions.
problem Understanding knotted and linked objects in higher dimensions.
method Using barbell diffeomorphisms to construct examples.
result Infinitely many knotted and linked objects in 4 and 5 dimensions.
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.