arXiv research

Minimal topology on surface homeomorphisms proven.

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.

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.

Paper compares topological and pro-étale fundamental groups.

For each $n\leq 6$, 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 $K_n$ in $S^3$.

The paper explores conditions for topological rigidity in quotients of the Davis complex.

We prove for the automorphism group of an arbitrary parabolic geometry that the $C^0$ and $C^{\infty}$ 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 $K_n$ by characterizing which $K_n$ can have a cyclic group, a dihedral group, or a subgroup of $D_m \times D_m$ where $m$ is odd, as its topological symmetry group.

This thesis introduces big mapping class groups and their structure.

The paper classifies topological holonomy groups in $SO(3)$.

Study proves topological complexity and LS-category inequalities for 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 $π_{1}^{top}$ 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.

Study index theory on Lie group homogeneous spaces using topological and analytic methods.

Authors compute fundamental groups for a specific topological group.

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.

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.

Study chaotic behavior in homeomorphism groups of countable products of spaces.

New topology shows Morse boundaries are topologically invariant.

Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.

Study the complexity of horizontality in 4-torus vector bundles.

Researchers found multiple surfaces with same topological and symmetry properties.

Topological normal generation proved for mapping class groups of certain surfaces.

We classify all groups which can occur as the topological symmetry group of some embedding of the Heawood graph in $S^3$.

This paper identifies all topological symmetry groups for Heawood family graphs.

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…

Researchers provide a simple topological method for Burau representations of loop braid groups.

We classify all groups which can occur as the orientation preserving topological symmetry group of some embedding of a Möbius ladder graph in $S^3$.

Study shows Hamiltonian diffeomorphisms form a connected component in $C^0$-topology for most symplectic rational surfaces.

Homology and cohomology theory for topological quandles computed.

Study calculates fundamental groups of torus knots using algebraic topology.

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.

Study on the topology of leaves in singular Riemannian foliations.

The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natu…

Involutions generate mapping class groups of infinite surfaces.

Surface groups are uniquely identified by their profinite completions.