arXiv research

Uniform interpretation of group theory in manifold homeomorphisms.

Proves existence of sentences to identify homeomorphic manifolds.

Study on homeomorphism groups of manifolds using set theory.

Study classifies 2-manifolds with special homeomorphism groups.

New homeomorphism found in Klein bottle group.

The study of periodic subgroups in homeomorphism groups of manifolds.

Study homeomorphism groups of ordinals, proving strong distortion and normal generators.

The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.

We extend the proof of automatic continuity for homeomorphism groups of manifolds to non-compact manifolds and manifolds with marked points and their mapping class groups. Specifically, we show that, for any manifold $M$ homeomorphic to the interior of a compact manifold, and a set $X \subset M$ homeomorphic to the uni…

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

The center of a quotient group of piecewise linear homeomorphisms is trivial.

Study shows certain surface homeomorphisms groups can't be precompact.

Classifies when homeomorphism groups of stable surfaces have automatic continuity.

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, the group of homeomorphisms of M …

Classifies Morse boundaries of 3-manifold groups.

Study shows certain mapping class groups cannot be realized as subgroup of homeomorphisms.

Study shows similar result to Margulis for Cantor set homeomorphisms.

The paper shows how certain groups' boundaries relate to the Sierpiński carpet.

The notion of a locally continuously perfect group is introduced and studied. This notion generalizes locally smoothly perfect groups introduced by Haller and Teichmann. Next, we prove that the path connected identity component of the group of all homeomorphisms of a manifold is locally continuously perfect. The case o…

Adapts pivoting technique to circle homeomorphisms for proofs.

We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condi…

The paper creates a deformation retraction for homeomorphisms of the projective plane.

We show that the group of all pl-homeomorphisms of the reals having bounded slopes surjects on the group $QI({\Bbb R})$ of all quasi-isometries of ${\Bbb R}$. We prove that the following groups can be imbedded in $QI({\Bbb R})$: The group of compactly supported pl-homeomorphisms of the reals, the Richard Thompson group…

The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.

Minimal topology on surface homeomorphisms proven.

The paper proves bounded cohomology properties of Euclidean space groups.

Study cohomology of homeomorphisms and diffeomorphisms of manifolds.

Finite groups act freely on surfaces but not on 3-manifolds.

Smooth actions of cyclic groups on 3-manifolds are conjugate to smooth ones.

An important theorem of Ling states that if $G$ is any factorizable non-fixing group of homeomorphisms of a paracompact space then its commutator subgroup $[G,G]$ is perfect. This paper is devoted to further studies on the algebraic structure (e.g. uniform perfectness, uniform simplicity) of $[G,G]$ and $[\tilde G,\til…

Study weak conjugacy in surface homeomorphisms.

Proofs non-realizability of mapping class group via homeomorphisms, resolves Thurston's conjecture.

We present a simple-to-apply criterion for recognizing topological groups that are (locally) homeomorphic to LF-spaces.

Automorphism group of nonorientable surface curve graph matches surface homeomorphisms.

We prove that for any non-compact connected surface $M$ the group $H_c(M)$ of compactly suported homeomorphisms of $M$ endowed with the Whitney topology is homeomorphic to $R^\infty\times l_2$ or $Z\times R^\infty\times l_2$.

In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under t…

For each $n > 0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations `conjugated by $z \to z^n$'. We show that these families are free of relations, which determines the structure of `the group of homeomorphisms of finite type'. We also discuss a numbe…

Suppose $X$ and $Y$ are compact connected topological 4-manifolds with fundamental group $π$. For any $r \geqslant 0$, $Y$ is $r$-stably homeomorphic to $X$ if $Y \# r(S^2 \times S^2)$ is homeomorphic to $X \# r(S^2\times S^2)$. How close is stable homeomorphism to homeomorphism? When the common fundamental group $π$ i…

Automorphisms of fine curve graph match surface homeomorphisms.

Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.

Gauge theory for families helps compare 4-manifold groups.

The paper studies quasimorphisms and distortion in homeomorphism groups of manifolds.

In this paper, we will show that the projection $\text{Homeo}^+(D^2_n)\to B_n$ does not have a section; i.e. the braid group $B_n$ cannot be geometrically realized as a group of homeomorphisms of a disk fixing the boundary point-wise and $n$ marked points in the interior as a set. We also give a new proof of a result o…

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.

We show that the topological groups $Diff_{+}^{1}(I)$ and $Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite to…

Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invari…

Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.

Study surfaces in 4-manifolds with cyclic fundamental group.