Minimal topology on surface homeomorphisms proven.
problem Proving the compact-open topology is minimal for surface homeomorphisms.
method Combining Hausdorff group topology properties and automatic continuity results.
result Compact-open topology is unique Hausdorff separable group topology on surface homeomorphisms.
The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
problem The minimality of compact-open topology on diffeomorphism and homeomorphism groups.
method Analyzing the compact-open topology on diffeomorphism and homeomorphism groups of smooth manifolds.
result The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
This paper classifies topological symmetry groups for Petersen family graphs.
problem Understanding symmetries of graphs embedded in 3D space.
method Examined all embeddings of Petersen family graphs in S3 and classified their topological symmetry groups. result Identified all possible groups that can be realized as topological symmetry groups for each graph in the Petersen family.
We characterize all groups which can occur as the topological symmetry group or the orientation preserving topological symmetry group of some embedding of the Petersen graph in S^3.
We give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Moreover, we investigate the profinite topology of the direct product of two free groups. We show that the profinite topology of the above group is strongly connected wit…
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
Paper compares topological and pro-étale fundamental groups.
problem No specific problem stated; comparing two fundamental groups.
method Constructs a comparison map between topological and pro-étale fundamental groups.
result Establishes a map between fundamental groups.
For each n≤6, we characterize all the groups which can occur as either the orientation preserving topological symmetry group or the topological symmetry group of some embedding of Kn in S3.
The paper explores conditions for topological rigidity in quotients of the Davis complex.
problem Understanding when quotients of the Davis complex are topologically rigid.
method Analyzing quotients of the Davis complex of right-angled Coxeter groups and conditions on defining graphs.
result Introduction of infinitely many infinite topologically rigid subclasses.
We prove for the automorphism group of an arbitrary parabolic geometry that the C0 and C∞ topologies coincide, and the group admits the structure of a Lie group in this topology. We further show that this automorphism group is closed in the homeomorphism group of the underlying manifold.
In this paper we complete the classification of topological symmetry groups for complete graphs Kn by characterizing which Kn can have a cyclic group, a dihedral group, or a subgroup of Dm×Dm where m is odd, as its topological symmetry group.
This thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
The paper classifies topological holonomy groups in SO(3).
problem Classifying holonomy groups in SO(3). method Analyzing and categorizing groups based on their properties.
result Different types of holonomy groups in SO(3) are identified and classified. Surveying matrix group actions on manifolds.
problem Topological Zimmer's conjecture on matrix group actions on manifolds.
method Surveying existing research and literature.
result Status update on matrix group actions on manifolds.
Study proves topological complexity and LS-category inequalities for specific groups and manifolds.
problem Proving inequalities for topological complexity and LS-category of specific groups and manifolds.
method Analyzing torsion free hyperbolic and nilpotent groups, lens spaces, using inequalities and counter-examples.
result Proves inequalities for topological complexity and LS-category of specific groups and manifolds.
The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. We completely describe the space of closed sugroups of the group RxZ, which is not trivial : for example, its fundamental group is uncountable.
The topological fundamental group π1top is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased space X, we compute the topological fundamental group of the "hoop earring" spac…
We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of the complete graph K_{4r+3} in S^3.
We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.
This paper determines all possible topological symmetry groups of generalized Petersen graphs.
problem Identifying all topological symmetry groups of generalized Petersen graphs.
method Analyzing embeddings of generalized Petersen graphs in S3 and considering homeomorphisms. result All groups that can be topological symmetry groups of generalized Petersen graphs are identified.
Study index theory on Lie group homogeneous spaces using topological and analytic methods.
problem Index theory on homogeneous spaces of Lie groups.
method Topological and analytic approaches: Riemann-Roch formula and heat kernel methods.
result Local index formula representing higher indices of equivariant elliptic operators.
Unified approach to homological representations of topological groups.
problem Constructing homological representations of topological groups.
method Functorial approach using topological enrichment of Quillen bracket construction.
result Unified construction of homological representations for mapping class groups and motion groups.
Authors compute fundamental groups for a specific topological group.
problem Computing fundamental groups of a specific topological group.
method Direct computation of fundamental groups.
result Fundamental groups π1Diff∂(D4k) for k≥3 are computed. A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to pr…
The study finds conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
problem Conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
method Analyzes conditions and examples to generalize results on topological complexity and Lusternik-Schnirelmann category.
result Generalizes results on topological complexity and Lusternik-Schnirelmann category for manifolds with abelian fundamental groups.
We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the group-compact coarse structure generalizing several familiar results for discrete group…
We construct an explicit topological model (similar to the topological Springer fibers appearing in work of Khovanov and Russell) for every two-row Springer fiber associated with the even orthogonal group and prove that the respective topological model is homeomorphic to its corresponding Springer fiber. This confirms …
For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks fa…
Overview of infinite surface mapping class groups.
problem Understanding mapping class groups of infinite surfaces.
method Survey of recent research findings.
result Recent developments in mapping class groups of infinite surfaces.
Study chaotic behavior in homeomorphism groups of countable products of spaces.
problem Investigate chaotic behavior in homeomorphism groups of countable products of various metrizable topological spaces.
method Construct numerous examples of chaotic groups of homeomorphisms of countable products of spaces.
result New chaotic groups of homeomorphisms of countable products of various metrizable topological spaces are discovered.
New topology shows Morse boundaries are topologically invariant.
problem Topological invariance of Morse boundaries in CAT(0) cubical groups.
method New hyperbolic topology induced by group actions on hyperbolic spaces.
result Sublinearly Morse boundaries are homeomorphic up to visual topology.
Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.
problem Minimal topological generating sets of mapping class groups consisting of torsion elements.
method Investigation of minimal topological generating sets for Map(S(n)) consisting entirely of torsion elements, with special attention to involutions. result Minimal topological generating sets for Map(S(n)) consisting of torsion elements are found for various n. Study the complexity of horizontality in 4-torus vector bundles.
problem Classify topological holonomy groups in SO(3).
method Analyze twistor spaces and oriented vector bundles over 2-torus.
result Discover many topological holonomy groups in SO(3) with noncommutative pairs.
Researchers found multiple surfaces with same topological and symmetry properties.
problem Determining unique free boundary minimal surfaces from topology and symmetry.
method Provided pairs of non-isometric surfaces with same genus, boundary components, and symmetry group.
result Infinitely many pairs of surfaces with same topology and symmetry group exist.
Topological normal generation proved for mapping class groups of certain surfaces.
problem Proving topological normal generation for mapping class groups of surfaces.
method Analyzing the end space of surfaces and using topological normal closure properties.
result Topological normal generation is equivalent to uniquely self-similar for surfaces with countable end space.
This paper identifies all topological symmetry groups for Heawood family graphs.
problem Understanding symmetries of spatial graphs in 3D space.
method Analyzing automorphisms of graphs embedded in S3. result All graphs in the Heawood family are intrinsically chiral.
We classify all groups which can occur as the topological symmetry group of some embedding of the Heawood graph in S3.
We prove that for every closed, connected, orientable, irreducible 3-manifold, there exists an alternating group A_n which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group G, there is an embedding Γ of some graph in a hyperbolic rational homology 3-sp…
Paper lifts Artin's representation to loop braid groups topologically.
problem Lifting Artin's representation to loop braid groups.
method Injection of loop braid group into automorphisms of π-module.
result Topological interpretation of the injection as an extension of Artin's and Dahm's representations.
Researchers provide a simple topological method for Burau representations of loop braid groups.
problem Constructing Burau representations of loop braid groups.
method Simple topological construction of the Burau representations.
result One of the representations is more subtle and topologically natural but not easily combinatorially obvious.
We classify all groups which can occur as the orientation preserving topological symmetry group of some embedding of a Möbius ladder graph in S3.
Study shows Hamiltonian diffeomorphisms form a connected component in C0-topology for most symplectic rational surfaces.
problem Understanding the C0-topology of symplectic diffeomorphisms on rational surfaces. method Combining techniques from symplectic mapping class groups and C0-symplectic topology, establishing C0-distance estimates. result Hamiltonian diffeomorphisms form a connected component in C0-topology for all but a few exceptions on rational surfaces. Homology and cohomology theory for topological quandles computed.
problem Computing invariants for knot diagrams using quandle cocycles.
method Introducing homology and cohomology theory for topological quandles, studying their relation to quandle groups, and using topological quandle cocycles to compute state sum invariants.
result State sum invariants computed using topological quandle cocycles.
Study calculates fundamental groups of torus knots using algebraic topology.
problem Calculating the fundamental group of torus knots.
method Algebraic topology and group theory.
result Computed fundamental groups of torus knots.
We prove that there does not exist any connected topological proper loop homeomorphic to a quasi-simple Lie group and having a compact Lie group as the group topologically generated by its left translations. Moreover, any connected topological loop homeomorphic to the 7-sphere and having a compact Lie group as the grou…
We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically mixing. Lastly, we prove that any continuous endomorphism on an abelian polish se…
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.
Study on the topology of leaves in singular Riemannian foliations.
problem Characterizing the topology of leaves in singular Riemannian foliations.
method Analyzing the fundamental groups and using nilpotent spaces.
result Leaves of singular Riemannian foliations are finitely covered by nilpotent spaces when M is simply connected.