Study on curved spheres' topology.
problem Topology of curved spheres.
method Gromov-Hausdorff metric on isometry classes.
result Properties of curved spheres' metric space.
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48 results for “Topological spheres”
Disk complexes show 3-sphere surfaces are topologically minimal.
problem Understanding minimal surfaces in 3-sphere topology.
method Analyzing disk complexes of genus >1 Heegaard surfaces.
result Genus >1 Heegaard surfaces have minimal index 2g-1.
Identifies submanifolds as topological spheres in hyperbolic space.
problem Characterizing submanifolds as topological spheres in hyperbolic space.
method Defines conditions on Ricci curvature and mean curvature vector length.
result Identifies submanifolds as topological spheres under given conditions.
Splitting spheres for -links in math.GT
The paper proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
problem Proving the existence of infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
method Establishing a general sufficient condition for infinitely many topologically non-isotopic splitting 3-spheres in connected sums of 4-manifolds.
result Proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
The study examines whether certain 4-manifolds can have flat embedded spheres.
problem Whether specific 4-manifolds can have smooth or topologically flat embedded 2-spheres.
method Analyzes the second homology of specific simply connected closed 4-manifolds.
result Determines conditions under which certain 4-manifolds can have flat embedded spheres.
Linking topological spheresmath.GT
No 3-sphere can be linked with a specific embedding of S^1 in R^5.
problem Linking problem between 3-spheres and a specific embedding of S^1 in R^5.
method Topological embedding and sphere removal analysis.
result No 3-sphere can be linked with the specific embedding of S^1 in R^5.
New groups found that act on spheres topologically but not smoothly.
problem Finding new groups that act on spheres topologically but not smoothly.
method Proving existence of a finite group G that acts topologically on S^d but not by orthogonal matrices.
result Existence of a finite group G that acts topologically on S^d but not by orthogonal matrices.
Hyperbolic groups act on spheres, and nearby actions are semi-conjugate.
problem Stability of group actions on spheres.
method Topological stability in dynamical sense.
result Nearby actions are semi-conjugate to the standard boundary action.
The study proves that certain hypersurfaces in elliptic space forms are topologically rigid.
problem Characterizing the topological structure of hypersurfaces in elliptic space forms.
method Proving topological rigidity under curvature constraints and using the Gauss map.
result Hypersurfaces in elliptic space forms are diffeomorphic to spheres or their quotients.
This work explores the topology of curves on a 3-sphere with positive curvature.
problem Understanding the topology of spaces of curves with positive curvature in the 3-sphere.
method Represented curves in the 3-sphere as pairs of curves on the 2-sphere with restrictions.
result Obtained partial information about the homotopy type of spaces of such curves.
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
Minimal volume invariants, topological sphere theorems and biorthogonal curvature on 4-manifoldsmath.DG
The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…
We use an idea of Wang and Yau to give a new definition of quasi-local mass for a topological sphere in an initial date set. The new definition modifies Brown-York's definition by using certain spinor norm as lapse function. And it requires mean curvature of the topological sphere satisfies apparent horizon conditions,…
New minimal surfaces in spheres with complex topologies from capillarity.
problem Constructing minimal surfaces in spheres with rich topologies.
method General construction of embedded minimal and constant mean curvature surfaces in using capillary hypersurfaces.
result Non-trivial sphere bundles over various base spaces, including Stiefel manifolds and complex quadrics.
New theorem links tropical phased matroids to higher-dimensional spheres.
problem Understanding topological properties of tropical phased matroids.
method Proving homeomorphism between topological order complex and a sphere.
result Topological order complex of tropical phased matroids is a -sphere.
We show that an -bridge sphere for the unknot is a topologically minimal surface of index at most .
Topology helps estimate chromatic numbers of random graphs on spheres.
problem Estimating chromatic numbers of random graphs on spheres.
method Topology, specifically connectivity of Lóvasz's neighborhood complex.
result Connectivity bound is useful in dimensions 1 and 2, but generally poor.
Classifies topological types of unions of 3-punctured spheres in hyperbolic 3-manifolds.
problem Classifying unions of 3-punctured spheres in hyperbolic 3-manifolds.
method Topological classification based on geometric properties of 3-punctured spheres.
result Special types of unions appear in only one or a few hyperbolic 3-manifolds.
Paper constructs first critical bridge spheres for nontrivial links.
problem Finding critical bridge spheres for nontrivial links.
method Equivalent combinatorial definition of critical surfaces.
result First known critical bridge spheres for nontrivial links.
The paper explores the topology and curvature of isoparametric families in spheres.
problem Investigating the topology and curvature of isoparametric families in spheres.
method The paper investigates the topology and curvature of isoparametric families in spheres using homotopy, homeomorphism, diffeomorphism types, parallelizability, and Lusternik-Schnirelmann category.
result The paper determines conditions for non-negative sectional curvatures and positive Ricci curvatures in isoparametric families.
Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
The Spheres of Solmath.DG
Study on Sol's metric spheres and their properties.
problem Characterizing metric spheres in Sol.
method Detailed analysis of cut locus and Riemannian exponential map.
result Metric spheres in Sol are topological spheres with precise singular points.
Topology classifies bipolar surfaces; they are not embedded.
problem Classifying bipolar surfaces in the 5-sphere.
method Topological classification and area bounds calculation.
result Bipolar surfaces are not embedded.
Proves spheres with bounded curvatures must contain a unit ball.
problem Proving spheres with bounded curvatures enclose a unit ball.
method Analyzing topological spheres in R^3 with bounded normal curvatures.
result Spheres with normal curvatures bounded by 1 must contain a unit ball.
Study submanifolds in spheres with Ricci curvature bounds.
problem Topology of submanifolds in spheres with Ricci curvature constraints.
method Investigates submanifolds in spheres with Ricci curvature lower bounds.
result Strong additional information on submanifold geometry.
Manifolds covered by two balls are homeomorphic to spheres.
problem Conditions for a manifold to be homeomorphic to a sphere using metric balls.
method Optimal geometric conditions for a Riemannian manifold covered by two metric balls.
result A Riemannian manifold covered by two metric balls is homeomorphic to a sphere.
Study shows equality in Hodge Laplacian bound occurs only on spheres.
problem Understanding when equality holds in Hodge Laplacian bounds for submanifolds.
method Analyzes closed submanifolds in space forms, proving equality on spheres.
result Equality in Hodge Laplacian bound occurs only on topological spheres.
This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
Study on compact hypersurfaces in spheres with Ricci curvature bounds.
problem Topology of compact hypersurfaces in spheres with Ricci curvature constraints.
method Use of Bochner technique for stronger results.
result Stronger results than previous studies in higher codimensions.
Study integral bounds for submanifolds in low codimension with topological implications.
problem Integral curvature bounds and topological obstructions for submanifolds.
method Integral curvature bounds in terms of Betti numbers, -pinched immersions, and pinched second fundamental form.
result Obtained topological obstructions for -pinched immersions and intrinsic obstructions for minimal submanifolds in spheres.
Study shows knots in homology spheres can be equivalent to knots in 3-sphere after any filtration step.
problem Detecting knots in homology spheres using the solvable filtration.
method Proved that for any knot in a homology sphere, there exists a knot in the 3-sphere equivalent modulo any term of the solvable filtration.
result Knots in homology spheres can be equivalent to knots in 3-sphere after any filtration step.
Simple formulas for sphere eversions using ruled surfaces.
problem Complex equations for sphere eversions.
method Explicit formulas for sphere and Boy surfaces using ruled surfaces.
result Visualizable formulas for sphere eversions.
This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. It presents the initial solutions of the fabled Double Suspension Conjecture. (The other two articles are: 'Approximating certain cell-like maps by homeomorphisms' and 'T…
New links split by integer homology spheres but not by others.
problem Characterizing links split by integer homology spheres.
method Constructing specific links and homology spheres.
result Infinite families of links and homology spheres split by specific ones but not by others.
Modified proof constructs dual spheres for 4-manifolds.
problem Prove dual spheres for 4-manifolds.
method Modified proof of disc embedding theorem, geometric construction.
result Constructs geometrically dual spheres.
The paper confirms conjectures about the topology of triangulated polyhedra and geodesic triangulations on spheres.
problem Topology of spaces of convex polyhedra and Delaunay triangulations on spheres.
method Variational principles on triangulated surfaces.
result Spaces of Delaunay triangulations have the same homotopy types as their smooth counterparts on the unit 2-sphere.
Proves constraints on groups extending Möbius transformations on spheres.
problem Constraints on groups extending Möbius transformations on spheres.
method Proved constraints through group transitivity and topological entropy analysis.
result Groups must be 4-transitive or arc 4-transitive, and contain elements of positive topological entropy.
New 4D examples show sphere equivalence doesn't imply isotopy.
problem Understanding when sphere equivalence implies topological isotopy in 4-manifolds.
method Constructing specific 4-manifolds with non-isotopic but equivalent 2-spheres.
result Gabai's 4D Lightbulb Theorem does not generalize to arbitrary 4-manifolds.
Authors prove a conjecture about the Goeritz group of 3-sphere Heegaard splittings.
problem The Goeritz group of a genus g Heegaard splitting of the 3-sphere.
method Proof relies on the topological minimality of Heegaard surfaces and their disk complexes.
result The Goeritz group is generated by four specific elements for g ≥ 3.
Invariants of -knotsmath.GT
Survey of invariants for knotted 2-spheres in 4-space.
problem Characterizing knotted 2-spheres in 4-dimensional space.
method Algebraic topology of knot exterior, gauge theory, combinatorial methods.
result Details are scarce and new results inexistent.
Proof of genus formula for 3-manifolds using arithmetic topology.
problem Proving a genus formula for finite abelian branched covers over integral homology 3-spheres.
method Utilizing analogies of arithmetic topology and Hasse norm principle.
result Presentation of an Iyanaga--Tamagawa type genus formula.
This paper formalizes the h-principle and sphere eversion in differential topology.
problem Formalizing the h-principle and sphere eversion in differential topology.
method Lean formalization of the local h-principle for first-order partial differential relations, using convex integration.
result Reproves Smale's sphere eversion theorem and formalizes advanced mathematics.
Linear Microbundlesmath.DG
Extends calculus to topological manifolds using generalized functions.
problem Proving the existence of non-singular generalized tangent vector fields on spheres.
method Develops a theory of generalized functions and applies it to continuous maps between topological spaces.
result Shows coherence between non-existence of smooth vector fields on spheres and existence of generalized ones.
The paper proves sphere theorems for submanifolds in Kähler manifolds.
problem Sphere theorems for submanifolds in Kähler manifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler manifolds, especially in complex space forms.
result Proves sphere theorems for submanifolds in Kähler manifolds.
The paper studies the topology of hyperbolic actions on manifolds, focusing on spheres and projective spaces.
problem Understanding the topology of hyperbolic actions on manifolds.
method Analyzing the -action generated by a fixed vector, studying the number of hyperbolic domains and fixed points, and constructing specific actions.
result Results on the number of hyperbolic domains and fixed points, and detailed analysis of 2-sphere cases.
The study classifies parallel mean curvature spheres in a sphere-hyperbolic product space.
problem Understanding surfaces with parallel mean curvature in a specific Riemannian product space.
method Analyzing the holomorphic quadratic differential and topological constraints.
result Classification of all parallel mean curvature spheres with vanishing differential.
Generating the Goeritz group of math.GT
New generators prove sufficiency for Goeritz group of 3-sphere.
problem Proving sufficiency of specific generators for Goeritz group.
method Expanding Powell's proposed generators to include all eyeglass twists and topological conjugates.
result Natural expansion of Powell's generators suffices to generate Goeritz group.
Sphere theorems for submanifolds in Kähler and Sasaki spaces.
problem Proving sphere theorems for Lagrangian and Legendrian submanifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler and Sasaki spaces.
result Proved sphere theorems for Lagrangian and Legendrian submanifolds.