Approaches 4D Schoenflies via pseudo-isotopy.
problem Smooth 4D Schoenflies conjecture.
method Pseudo-isotopy theory.
result Offers new approach to 4D Schoenflies.
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48 results for “Schoenflies spheres”
New examples of Schoenflies balls are produced using a 5D approach.
problem Identifying Schoenflies balls that are not standard.
method Using a 5-dimensional perspective, algebraic and geometric handle cancellation.
result New examples of Schoenflies balls not known to be standard are produced.
3D Schoenflies theorem for simply-connected 2-complexes.
problem Embedding simply-connected 2-complexes in 3-space uniquely.
method Proving a 3-dimensional Schoenflies theorem for 3-connected link graphs.
result Essentially unique locally flat embedding into 3-sphere.
The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.
problem Understanding the homotopy types of diffeomorphism groups and sphere embeddings.
method Cerf's upgraded proof, scanning maps, canceling handles, Embedding Calculus.
result The monoid of Schoenflies spheres forms a group under connect-sum.
In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…
The stable Andrews-Curtis conjecture in combinatorial group theory is the statement that every balanced presentation of the trivial group can be simplified to the trivial form by elementary moves corresponding to "handle-slides" together with "stabilization" moves. Schoenflies conjecture is the statement that the compl…
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.
There is a relation between the generalized Property R Conjecture and the Schoenflies Conjecture that suggests a new line of attack on the latter. The approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices to prove the genus 3 case, even in the absence of new progress on the generalized Propert…
We show that both, the Jordan curve theorem and the Schoenflies theorem extend to non-metric manifolds (at least in the two-dimensional context), and conclude by some dynamical applications à la Poincaré-Bendixson.
BiLipschitz mappings can be extended to preserve area.
problem Extending biLipschitz mappings to preserve area.
method Proving biLipschitz mappings can be extended to biLipschitz mappings preserving area.
result BiLipschitz mappings can be extended to preserve area.
This paper gives a complete proof of the result announced in the title.
We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that the intersection of A and B is H and the union is G and such that the complementary graphs A,B are both d-balls. The graph theoretic de…
This paper describes a method to construct standard 4-balls from homotopy 4-balls in .
problem The problem is whether every homotopy 4-ball in is standard.
method The approach is to use Stein surfaces and pseudoconvex domains to construct a diffeomorphic domain that is the union of three pseudoconvex domains, ensuring it is a standard 4-ball.
result The construction method ensures that the domain is a standard 4-ball, providing a compelling reimbedding construction for homotopy 4-balls in .
This is the announcement of an alternative approach to the 3-dimensional Poincaré Conjecture, different from Perelman's big and spectacular breakthrough. No claim concerning the other parts of the Thurston Geometrization Conjecture, come with our purely 4-dimensional line of argument.
Low-entropy surfaces can be flowed into spheres and cylinders.
problem Proving mean curvature flow for low-entropy hypersurfaces.
method Low-entropy density drop argument and recent work on hypersurfaces.
result Closed hypersurfaces with entropy ≤ 2 can be flowed into spherical and cylindrical shapes.
Involutions on S^4math.GT
The paper classifies involutions on S^4, proving linearities under certain conditions.
problem Classifying involutions on S^4 with specific fixed-point sets.
method Combining surgery theory, Schoenflies theorem, and equivariant topology.
result Linear involutions on S^4 with 1-dimensional fixed-point sets are proven.
This is an expository paper giving a proof of the existence and uniqueness of smooth structures (hence also PL structures) on topological surfaces. Most published proofs rely on the topological Schoenflies theorem, but here we use instead the Kirby torus trick. This has the advantage of reducing the point-set topology …
We give a general treatment of the somewhat unfamiliar operation on manifolds called Connected Sum at Infinity, or CSI for short. A driving ambition has been to make the geometry behind the well definition and basic properties of CSI as clear and elementary as possible. CSI then yields a very natural and elementary pro…
We show that a smooth embedding of a closed 3-manifold in S^3 x R can be isotoped so that every generic level divides S^3 x t into two handlebodies (i.e., is Heegaard) provided the original embedding has a unique local maximum with respect to the R coordinate. This allows uniqueness of embeddings to be studied via the …
Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.
problem Approximating weak mean curvature flows with singularities using smooth flows.
method Combining Choi-Haslhofer-Hershkovits and Choi-Haslhofer-Hershkovits-White work on canonical neighbourhoods and barriers to flows with surgery.
result Smooth flows with surgery can approximate weak mean curvature flows with spherical and neck-pinch singularities.
In a recent paper, {\it Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (II)}, we discussed two algorithms for deforming and contracting a simply connected discrete closed manifold into a discrete sphere. The first algorithm was a continuation of work that began in {\it Algorithms fo…
Classifies manifolds and discrete subgroups of Lie groups using descriptive set theory.
problem Classifying manifolds and discrete subgroups of Lie groups.
method Descriptive set theory and Borel complexity computations.
result Complexity of homeomorphism problems for manifolds and conjugacy relations for subgroups.
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.
Smooth homotopy 4-spheremath.GT
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.
Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexesmath.GT
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.
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.
The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.
problem Proving existence of minimal 2-spheres or optimal foliations in arbitrary Riemannian 3-spheres.
method Analyzing the properties of arbitrary Riemannian metrics on 3-spheres.
result The existence of at least two minimal 2-spheres or an optimal foliation in 3-spheres with arbitrary metrics.
Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.
problem Existence of pairwise non-isotopic splitting spheres for unlinked 2-spheres in 4-space.
method Analytical proof showing non-isotopic spheres.
result Infinitely many non-isotopic splitting spheres found.
Reduces weak reducing pairs to spheres in 3-sphere Heegaard surfaces.
problem Finding reducing spheres for weak reducing pairs in Heegaard surfaces.
method Proves existence of reducing spheres for weak reducing pairs in 3-sphere Heegaard surfaces.
result Reduction of weak reducing pairs to spheres if genus is at most 3.
The study shows how to construct -spheres from -spheres and -balls without additional vertices.
problem Constructing -spheres from -spheres and -balls without additional vertices.
method Examining specific types of spheres (flag, stacked, join of spheres) and -balls to determine if constructions can be made without extra vertices.
result Affirmative answers to constructing -spheres from -spheres and -balls without additional vertices for certain types of spheres and -balls.
Patterns and Tracksmath.GT
New proof for sphere recognition algorithm.
problem Sphere recognition algorithm proof.
method New proof of a lemma in Abigail Thompson's algorithm.
result New proof of a lemma in Abigail Thompson's proof of the Recognition Algorithm for 3-spheres.
Stability of Convex Spheresmath.DG
Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
Soliton Spheresmath.DG
Soliton spheres are immersed 2-spheres in the conformal 4-sphere S^4=HP^1 that allow rational, conformal parametrizations f:CP^1->HP^1 obtained via twistor projection and dualization from rational curves in CP^{2n+1}. Soliton spheres can be characterized as the case of equality in the quaternionic Pluecker estimate. A …
Study on sphere immersions and their stability indices.
problem Analyzing the stability of sphere immersions.
method Calculation of Morse indices and stability indices for specific sphere immersions.
result Bounds on stability index of associative cone in .
The paper constructs biharmonic maps between spheres using polynomial maps.
problem Creating biharmonic maps between spheres.
method Using harmonic homogeneous polynomial maps of different degrees to generate proper biharmonic maps.
result Established a method for constructing proper biharmonic product maps.
Classification of constant curvature surfaces in Berger spheres.
problem Identifying complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres.
method Complete classification through detailed analysis of Clifford tori and spheres.
result Rotationally invariant spheres with constant Gauss curvature are the only topological spheres in Berger spheres for .
Characterizes a specific type of convex curves on a 3-sphere.
problem Understanding convex curves on a 3-sphere.
method Decomposes curves on 3-sphere into 2-sphere curves, characterizes locally convex ones.
result Completely characterized a class of convex curves on the 3-sphere.
Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.
problem Connectivity problem in reducing sphere complex for genus-4 Heegaard surface.
method Presented a sufficient condition for a non-separating weak reducing pair to be separated by a reducing sphere.
result Reduced connectivity problem to showing disjointness of representative reducing spheres from a fixed disk.
Paper studies inscribed sphere and lines through centers of Apollonius spheres in n dimensions.
problem Tangency of spheres and lines through their centers.
method Lie sphere geometry and two-step construction of Apollonius spheres.
result Center of inscribed sphere coincides with point .
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.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mea…
We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds , where or , which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…
In this paper we show that for m>n the set of cobordism classes of maps from m-sphere to n-sphere is trivial. The determination of the cobordism homotopy groups of spheres admits applications to the covers for spheres.
Study finds infinite families of Sasaki-Einstein metrics on spheres.
problem Finding Sasaki-Einstein metrics on spheres and exotic spheres.
method Analyzing odd-dimensional spheres and exotic spheres that bound parallelizable manifolds.
result Infinitely many families of Sasaki-Einstein metrics on spheres and exotic spheres.
We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum…
We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definitions of spheres in graph theory. We also reformulate Morse conditions in terms of the center manifolds, the level surface graphs {f=f(x)} in the unit sphere S(x). In the Morse case these graphs are either spheres, the empty…
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.