Study on 3-sphere Goeritz group's twisted first homology group.
problem Determining the twisted first homology group of a specific Goeritz group.
method Defined genus-g Goeritz group, analyzed 3-sphere with genus-2 Heegaard splitting.
result Determined the twisted first homology group of the genus-2 Goeritz group.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
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…
Study connects Morin singularities to sphere homotopy groups.
problem Computing stable homotopy groups of spheres.
method Apply Morin singularities to sphere homotopy groups.
result Differentials in spectral sequence linked to sphere homotopy groups.
Study on group actions on spheres using multisymplectic geometry.
problem Existence of homotopy comoment maps for compact Lie group actions on spheres.
method Investigation of multisymplectic actions and comoments on spheres.
result Explicit constructions of comoments for interesting cases.
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.
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…
Study excludes smooth actions on low-dimensional spheres for certain finite Oliver groups.
problem Excluding smooth effective one fixed point actions of finite Oliver groups on low-dimensional spheres.
method Strategy based on Oliver groups and Laitinen-Morimoto-Pawałowski results.
result Nonexistence of smooth effective one fixed point actions for specified finite Oliver groups.
Study of CMC spheres in Heisenberg group, proving conjecture.
problem Proving conjecture about CMC spheres in Heisenberg group.
method Analyzing properties of spheres in Riemannian Heisenberg group H1. result Supporting conjecture that CMC spheres are isoperimetric sets.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
Study connects Morin singularities to sphere homotopy groups.
problem Understanding stable homotopy groups of spheres.
method Establishes a connection between Morin singularities and sphere homotopy groups.
result Describes behavior of singularity strata images around more complex strata.
A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that…
Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
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.
Study finite groups acting on 3D shapes with rational homology.
problem Identify finite groups that can act freely on rational homology 3-spheres.
method Cohomology of groups to analyze finite group actions.
result Characterize finite groups that can act freely and trivially on rational homology 3-spheres.
Researchers compute TQFT representation for sphere with 4 punctures.
problem Computing the representation of mapping class group for a sphere with 4 punctures.
method Non semi-simple TQFT approach, focusing on sphere with 4 punctures.
result The representation is faithful and compared with braid groups.
Study on group cocycles for volume-preserving diffeomorphisms.
problem Understanding group cocycles on volume-preserving diffeomorphisms.
method Constructed two types of group cocycles on the volume-preserving diffeomorphism group.
result One cocycle yields the Euler class of flat sphere bundles for the sphere.
Study Goeritz groups of link decompositions, focusing on their asymptotic behavior.
problem Understanding the asymptotic behavior of Goeritz groups for link decompositions.
method Defined Goeritz groups for link decompositions, analyzed their properties, and discussed their asymptotic behavior.
result Discussed the asymptotic behavior of minimal pseudo-Anosov entropies and related it to Goeritz groups of Heegaard splittings.
It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each …
New rack and multiple group rack cohomology for surfaces in 3-sphere.
problem Categorizing compact oriented surfaces in 3-sphere based on symmetry.
method Developed cohomology theory for racks and multiple group racks, constructed cocycle invariants.
result Identified new symmetry types of surfaces in 3-sphere.
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.
New homotopy spheres help solve complex manifold problems.
problem Constructing and characterizing exotic manifolds.
method Introducing Farrell-Jones spheres and studying their properties.
result Farrell-Jones spheres provide examples of exotic manifolds.
Contact group retracts to unitary subgroup.
problem Understanding contact structures on 3-sphere.
method Proving deformation retraction to unitary subgroup.
result Group of contactomorphisms retracts to U(2).
Sphere bundles with 1/4-pinched metrics are induced by vector bundles.
problem Existence of 1/4-pinched metrics on sphere bundles.
method Proving all smooth sphere bundles with 1/4-pinched fiberwise metrics are induced bundles of vector bundles.
result Existence of many smooth n-sphere bundles without 1/4-pinched positively curved metrics.
Loops on spheres with compact-free inner mapping group are homeomorphic to the circle.
problem Characterizing loops on spheres with specific inner mapping groups.
method Topological and differential analysis of loops using Lie groups and Fourier series.
result Loops on spheres with compact-free inner mapping group are homeomorphic to the circle.
Study automorphisms of pure braid groups on sphere homotopy groups.
problem Understanding automorphisms' effect on sphere homotopy groups.
method Examined Delta-group structure, proved invariance of cycle and boundary groups, computed action for few strands.
result Induced action of all automorphisms of pure braid groups on sphere homotopy groups.
Study invariants of Z/p-homology 3-spheres from abelianization of mapping class groups.
problem Deciding and constructing invariants of Z/p-homology 3-spheres. method Formulating a criterion and using families of trivial 2-cocycles on the abelianization of the level-p mapping class group. result Disproved a conjectured extension of the Casson invariant for rational homology 3-spheres.
Study extends symmetries of sphere points to surface mapping classes.
problem Understanding symmetries of points on spheres and their connections.
method Establishing isomorphisms between moduli spaces and mapping class groups.
result Finitely many integral braid group orbits in rank 4 Stokes matrices.
Researchers describe finite orbits in character varieties of a sphere.
problem Characterizing finite orbits in character varieties of the punctured sphere.
method Coalescence procedure and theory of finite complex reflection groups.
result Complete description of finite braid group orbits in Aff(C)-character varieties.
The paper proves group actions on spheres with odd fixed points.
problem Finite group actions on homology six-spheres with odd Euler characteristics.
method Analyzes smooth actions and fixed point sets of finite groups.
result The group is one of three specific types, and the fixed point set is a single point.
The study examines the regularity of spheres in homogeneous groups using specific distance criteria.
problem Understanding the regularity of metric spheres in homogeneous groups.
method Investigation of left-invariant distances on Lie groups with homothetic automorphisms, focusing on Carnot groups and Heisenberg group.
result Established criteria for the regularity of metric spheres in homogeneous groups, including the Heisenberg group.
Simplified proof of sphere group classification.
problem Classifying finite groups acting on spheres.
method Simplified proof and removal of redundancy.
result Improved classification of sphere groups.
In Heisenberg group, bisectors are spinal spheres with specific curvature.
problem Understanding bisectors in the Heisenberg group.
method Showed bisectors are spinal spheres and calculated their curvature.
result Metric bisectors in Heisenberg group are spinal spheres with specific curvature.
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.
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.
Study Dehn-Seidel twists on Lagrangian spheres in K3 surfaces.
problem Understanding configurations of Lagrangian spheres in symplectic K3 surfaces.
method Use Seiberg-Witten theory and tools from symplectic mapping class groups.
result Proves algebraic independence of certain twists and generation results.
This is a survey on old and new results as well as an introduction to various related basic notions and concepts, based on two talks given at the International Workshop on Geometry and Analysis in Kemerovo (Sobolev Institute of Mathematics, Kemerovo State University) and at the University of Krasnojarsk in June 2011. W…
Paper disproves a Smith conjecture about sphere actions.
problem Smith conjecture about sphere actions with two fixed points.
method Induction of group representations to show counterexamples.
result Negative answer to Smith conjecture for specific groups.
In this paper, we study the structure of homogeneous subgroups of the homeomorphism group of the sphere, which are defined as closed groups of homeomorphisms of the sphere that contain the rotation group. We prove two structure theorems about the behaviour and properties of such groups and present a diagram of the stru…
Classifies finite orbits of mapping class group action on character varieties.
problem Classifying finite orbits of mapping class group action on character varieties of punctured spheres.
method Inductive proof using Lisovyy--Tykhyy's classification for 4-punctured spheres as base case.
result Proves no finite orbits for 7-punctured spheres and unique 1-parameter family for 6-punctured spheres.
Study shows a copy of 7D real projective space in the space of almost complex structures on 6-sphere.
problem Understanding the topology of almost complex structures on the 6-sphere.
method Analyzes the fundamental and rational homotopy groups, computes homotopy fiber and groups, and generalizes to 6-manifolds.
result Induces isomorphism on fundamental and rational homotopy groups of the 6-sphere.
Classifies central extensions for area-preserving diffeomorphisms and shows they are fuzzy sphere limits.
problem Classifying central extensions for area-preserving diffeomorphisms.
method Classifying central extensions and showing they are fuzzy sphere limits of Kac-Moody cocycles.
result Central extensions are fuzzy sphere limits of Kac-Moody cocycles for large k.
Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.
problem Proving the uniqueness of Rohlin invariant and extending homology sphere invariants.
method Using the Rohlin invariant's uniqueness, the paper extends invariants from trivial 2-cocycles to those with 2-torsion.
result Generalized invariants of homology spheres with 2-torsion values.
Constructs chiral rational homology spheres with hyperbolic groups.
problem Existence of strongly chiral rational homology spheres with hyperbolic fundamental groups.
method Construction of rational homology spheres using r-spins and investigation of self-map degrees. result Strongly chiral rational homology spheres with hyperbolic fundamental groups constructed.
Study two homomorphisms to rational homology sphere group, proving new results on knot concordance.
problem Understanding knot concordance and homology sphere groups.
method Analyzing homomorphisms and using Tristram-Levine signatures.
result Knot concordant to a knot with determinant 1 if and only if it is smoothly slice.
New optimal surfaces found in Heisenberg group defy Euclidean sphere optimality.
problem Optimizing mean curvature in Heisenberg group sub-Riemannian setting.
method Developed variational theory, established first and second variation formulas, introduced new critical surfaces.
result Identified and characterized a new family of rotationally invariant critical surfaces, the Pansu-Minkowski spheres.
Locally flat 2-spheres in CP2 with knot group Z2 are ambiently isotopic if homologous.
problem Determining when locally flat 2-spheres in CP2 are ambiently isotopic. method Using knot groups and homology, proving isotopy based on homology equivalence and combining with previous results.
result Locally flat 2-spheres in CP2 with knot group Z2 are ambiently isotopic if they are homologous. R. S. Kulkarni showed that a finite group acting pseudofreely, but not freely, preserving orientation, on an even-dimensional sphere (or suitable sphere-like space) is either a periodic group acting semifreely with two fixed points, a dihedral group acting with three singular orbits, or one of the polyhedral groups, oc…