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.
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.
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.
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,…
We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also…
We show that the disk complex of a genus g>1 Heegaard surface for the 3-sphere is homotopy equivalent to a wedge of (2g−2)-dimensional spheres. This implies that genus g>1 Heegaard surfaces for the 3-sphere are topologically minimal with index 2g−1.
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 Sn 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 (2n−3)-sphere. We show that an (n+1)-bridge sphere for the unknot is a topologically minimal surface of index at most n.
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.
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.
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.
In this note we study whether specific elements in the second homology of specific simply connected closed 4-manifolds can be represented by smooth or topologically flat embedded spheres.
There is a topological embedding ι:S1→R5 such that π3(R5∖ι(S1))=0. Therefore, no 3-sphere can be linked with ι(S1).
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.
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…
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.
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.
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.
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.
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.
In this paper, we prove some differentiable sphere theorems and topological sphere theorems for submanifolds in Kähler manifold, especially in complex space forms.
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.
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.
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.
In this paper, we prove some differentiable sphere theorems and topological sphere theorems for Lagrangian submanifolds in Kähler manifold and Legendrian submanifolds in Sasaki space form.
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.
The standard actions of finite groups on spheres S^d are linear actions, i.e. by finite subgroups of the orthogonal group O(d+1). We prove that, in each dimension d>5, there is a finite group G which admits a faithful, topological action on a sphere S^d but is not isomorphic to a subgroup of O(d+1). The situation remai…
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.
We construct infinitely many smooth oriented 4-manifolds containing pairs of homotopic, smoothly embedded 2-spheres that are not topologically isotopic, but that are equivalent by an ambient diffeomorphism inducing the identity on homology. These examples show that Gabai's recent "Generalized" 4D Lightbulb Theorem does…
We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology…
The study shows manifolds with special generic maps also have nice multisections.
problem Characterizing manifolds with special generic maps and multisections.
method Analyzing manifolds with special generic maps and their properties, and showing how these maps restrict the differentiable structures of spheres and manifolds.
result Manifolds admitting special generic maps also admit nice generalized multisections.
The study connects projective codes to the distribution of zeros of odd maps.
problem Understanding the distribution of zeros of odd maps from spheres to Euclidean space.
method Using the topology of the space of probability measures on the sphere.
result Generalization of the Borsuk-Ulam theorem and its four consequences.
We provide a simple topological derivation of a formula for the Reidemeister and the analityc torsion of spheres.
In this note we provide natural optimal geometric conditions for a Riemannian manifold suitably covered by two open metric balls to be homeomorphic to a sphere. This can be viewed as a geometric analogue of Brown's theorem in topology stating that a closed manifold covered by two topological balls is a sphere.
A (positive) locally convex curve in the 2-sphere is a curve with positive geodesic curvature (i.e., which always turns left). In the 3-sphere, it is a curve with positive torsion. In this work we discussed the topology of spaces of such curves with prescribed initial and final jets. The case of the 2-sphere is underst…
New invariant computed for Brieskorn homology spheres.
problem Computing an invariant for specific manifolds.
method Using Dijkgraaf--Witten invariants in topological K-theory.
result Computed invariant for Brieskorn homology spheres.
New examples of knotting phenomena in 4-manifolds with specific fundamental groups.
problem Finding 2-spheres in simply connected 4-manifolds with prescribed fundamental groups.
method Construction of infinite sets of pairwise smoothly inequivalent 2-spheres that are topologically isotopic.
result First known examples of knotting phenomena in 4-manifolds with specific properties.
Abstract: Study gyrations of sphere products and connected sums, generalizing Fico's Lemmata.
problem Understanding the homotopy type of gyrations of sphere products and connected sums.
method Recasting Fico's Lemmata into modern homotopy theoretic setting.
result Generalization of Fico's Lemmata to sphere products and connected sums.
We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved 2-spheres.
Classifies Morse flows on 3-sphere with specific saddle connections.
problem Classifying Morse-Smale flows on a 3-sphere with specific saddle connections.
method Used generalized Heegaard diagrams (Pr-diagrams) to classify flows.
result Found all possible, up to homeomorphism, ways to embed two circles in a 2-sphere with no more than 10 points of transversal intersection.