The study connects spheres in specific surface curve graphs, proving connectivity and classifying components.
problem Proving connectivity and classifying components of spheres in curve graphs of low and medium complexity surfaces.
method Analyzing specific surfaces Σ2,0,Σ1,3,Σ0,6 and Σ0,5,Σ1,2, proving connectivity and classifying components. result Spheres of any radius are connected in Σ2,0,Σ1,3,Σ0,6, and the union of two consecutive spheres is connected in Σ0,5 and Σ1,2. 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.
Spheres in curve complexes are almost simply connected.
problem Understanding connectivity of spheres in curve complexes.
method Defining spheres as induced subgraphs and showing almost simple connectivity.
result Spheres in high-complexity surfaces are almost simply connected.
The invariant metric affine connections on Berger spheres which are Einstein with skew torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. In particular, every Berger sphere with Lorentzian signature admits invariant metric affine connections Einstei…
Spheres in curve graphs are connected, proving Gromov boundary linearity.
problem Understanding connectivity in curve graphs and their boundaries.
method Defining spheres and analyzing their connectivity for different complexities.
result Spheres in high complexity curve graphs are always connected, with weaker results for low complexity.
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.
A special knot in the Poincaré sphere leads to a unique connected sum of lens spaces.
problem Understanding the unique connected sum of lens spaces formed by a special knot in the Poincaré sphere.
method Analyzing the Seifert fibering and Dehn surgery of the Poincaré homology sphere.
result The only knot in the Poincaré sphere with a surgery to a connected sum of more than two lens spaces is the one mentioned.
We completely determine which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.
Bounds on saddle connections on flat spheres with conical singularities.
problem Counting saddle connections on flat spheres with conical singularities.
method Geometry of immersed disks and explicit upper bounds.
result Explicit upper bounds on the number and lengths of saddle connections.
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.
We classify the radially symmetric connections in vector bundles over round spheres by proving that they are all parallel.
In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit…
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. Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
problem Characterizing manifolds that are both rational homology spheres and double disk bundles.
method Analyzing the structure of manifolds as unions of disk bundles and using properties of rational homology and cohomology.
result Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
For each rational homology 3-sphere Y which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to Y but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,…
Proves a connectivity conjecture for free groups, showing homotopy type of spheres.
problem Establishing a connectivity conjecture for free groups.
method Provided homotopy-equivalent models of the common basis complex using free factors and sphere systems.
result The common basis complex of a free group of rank n has the homotopy type of a wedge of spheres of dimension 2n-3.
Any two homologous surfaces of the same genus embedded in a smooth 4-manifold X with simply-connected complements are shown to be smoothly isotopic in the connected sum of X and the product of a 2-sphere with itself, if the surfaces are ordinary, and in the connected sum of X with the non-trivial sphere bundle over the…
It is well-known that the Pachner graph of n-vertex triangulated 2-spheres is connected, i.e., each pair of n-vertex triangulated 2-spheres can be turned into each other by a sequence of edge flips for each n≥4. In this article, we study various induced subgraphs of this graph. In particular, we prove tha…
Study on negative Sasakian structures on specific 5-manifolds.
problem Existence of negative Sasakian structures on simply-connected 5-manifolds.
method Analysis of rational homology spheres and Smale-Barden manifolds.
result Proves existence of negative Sasakian structures on specific 5-manifolds.
We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection…
We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the …
The article provides formulas for the number of terms in connected sums of sphere products associated with dual-neighborly polytopes.
problem Understanding the number of terms in the connected sums of sphere products associated with dual-neighborly polytopes.
method Combinatorial operations and formulas for the number of terms in the connected sums of sphere products.
result Formulas for the number of terms in the connected sums of sphere products associated with dual-neighborly polytopes.
Anti-symplectic involutions connect a sphere in a symplectic surface.
problem Understanding involutions on Lagrangian spheres in symplectic quadrics.
method Using Hamiltonian isotopy to show connections between involutions.
result Anti-symplectic involutions are Hamiltonian isotopic.
Genus 3 Heegaard groups of lens space connected sums are finitely generated.
problem Understanding the structure of mapping class groups of genus 3 Heegaard splittings.
method Proved finitely generated property through connected reducing sphere complexes.
result Mapping class groups are finitely generated and complexes are connected.
Classifies exceptional Legendrian realizations of Hopf link connected sums.
problem Classifying exceptional Legendrian realizations of Hopf link connected sums.
method Complete coarse classification using Legendrian knot theory.
result First classification result about exceptional Legendrian representatives for Hopf link connected sums.
Morse theory connects low energy submanifolds in 3-sphere.
problem Understanding low energy submanifolds in the 3-sphere.
method Morse-theoretic techniques and negative gradient flow.
result Constructs connections between low energy critical submanifolds.
Study eternal solutions to Allen-Cahn equation on 3-sphere, connecting Clifford tori to equatorial spheres.
problem Understanding eternal solutions to the Allen-Cahn equation on the 3-sphere.
method Realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, using classifications and rigidity results.
result Construction of eternal integral Brakke flows connecting Clifford tori to equatorial spheres.
We show that if S is a finite type orientable surface of negative Euler characteristic which is not the 3-holed sphere, 4-holed sphere or 1-holed torus, then the ending lamination space of S is connected, locally path connected and cyclic.
A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann-Cartan space is introduced in a natura…
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.
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.
Researchers prove unique connection and curvature for Podleś quantum sphere.
problem Calculating curvature and Weitzenbock formula for Podleś quantum sphere.
method Using spectral triple and Dabrowski-Sitarz framework, they computed curvature tensors and proved a generalized Weitzenbock formula.
result The scalar curvature of Podleś sphere converges to 2 as q approaches 1.
We establish a connection between Morin singularities and stable homotopy groups of spheres. This connection allows us to describe how the images of singularity strata behave around the image of a more complicated stratum.
Confirming the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
problem Proving the finitely generated nature of the Goeritz group for genus-3 Heegaard splittings of the 3-sphere.
method Establishing the connectivity of reducing sphere complexes for the genus-3 case.
result Confirmation of the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
The paper constructs four-manifolds with lens space boundaries and explores sphere configurations in #nCP2.
problem Exploring configurations of spheres in #nCP2 and constructing four-manifolds with specific properties. method Constructing examples of simply connected four-manifolds with lens space boundaries using sphere plumbings in connected sums of CP2. result Examples of four-manifolds with lens space boundaries and configurations of spheres with self-intersection number 20.
Here we discuss an example of topologically isotopic but smoothly non-isotopic pair of 2-spheres in a simply connected 4-manifold, which become smoothly isotopic after stabilizing by connected summing with S^2 x S^2.
Classifies two families of simply connected 7-manifolds with minimal homological complexity.
problem Classifying simply connected rational homology 7-spheres that are not 2-connected.
method Complete classification of two families of manifolds using Milnor's λ-invariant and Eells-Kuiper μ-invariant.
result Minimal homological complexity among simply connected rational homology 7-spheres that are not 2-connected.
The paper classifies smooth structures on product manifolds of 3-connected 8-manifolds with spheres.
problem Classifying smooth structures on product manifolds.
method Computational and classification methods for concordance and diffeomorphism.
result Diffeomorphism classification of MimesS1 for specific M and k. 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.
The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. …
New geometric interpretation of discrete Willmore energy using rolling spheres connection.
problem Discrete formulation of Willmore energy for simplicial surfaces.
method Geometric interpretation of Möbius invariant discrete Willmore energy using rolling spheres connection.
result Clear geometric interpretations of discrete Willmore energy with manifest Möbius invariance.
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…
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.
Study Brieskorn spheres using Floer homology, generating infinite rank summands in homology cobordism.
problem Computing Heegaard Floer homologies of Brieskorn spheres.
method Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong.
result Brieskorn spheres generate infinite rank summands in the homology cobordism group.
Paper confirms Whitehead's conjecture for aspherical 2-complexes.
problem Whitehead's conjecture about aspherical 2-complexes.
method Argument on ribbon sphere-links, generalized for aspherical 2-complexes.
result Whitehead's conjecture confirmed for aspherical 2-complexes.
Paper proves no stable Yang-Mills fields on spheres.
problem Existence of stable Yang-Mills fields on spheres.
method Analyzes C2 neighborhoods of Euclidean sphere metrics and warped product manifolds. result No nontrivial weakly stable Yang-Mills connections in specified neighborhoods.
New Sasaki-Einstein 7-manifolds found, including rational homology 7-spheres and connected sums.
problem Finding new Sasaki-Einstein 7-manifolds and understanding their properties.
method Calculating homology groups of specific 7-manifolds using Thom-Sebastiani sums and quasi-regular metrics.
result 52 new Sasaki-Einstein rational homology 7-spheres and 124 new 2-connected 7-manifolds homeomorphic to S3imesS4 were found. It is one of the most important facts in 4-dimensional topology that not every spherical homology class of a 4-manifold can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a sphere vanish if one modifies the …