arXiv research

Study covers of sphere with homeomorphisms lifting property.

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standar…

We characterize the cyclic branched covers of the 2-sphere where every homeomorphism of the sphere lifts to a homeomorphism of the covering surface. This answers a question that appeared in an early version of the erratum of Birman and Hilden [2].

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

Alexander trick applied to homology spheres for manifold homeomorphisms.

The paper proves manifolds homeomorphic to spheres under specific Morse-Bott conditions.

CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.

Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.

Every countable compact subset of sphere is tame.

New theorem links tropical phased matroids to higher-dimensional spheres.

Gluck twists on spheres yield equivalent 4-manifolds under certain conditions.

We prove that any topological loop homeomorphic to a sphere or to a real projective space and having a compact-free Lie group as the inner mapping group is homeomorphic to the circle. Moreover, we classify the differentiable $1$-dimensional compact loops explicitly using the theory of Fourier series.

Cyclic covers of knots uniquely determine the original knot.

Alexandrov spaces have a special stratification that maps to spheres.

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m>4) of positive section…

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…

The purpose of this paper is to study how small orbits of periodic homemorphisms of spheres can be.

The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a constant p(n)>0 such that (1) If M^{2n} admits an effective isometric \Bbb Z_p^k-act…

We discuss the concept of the shadow boundary of a centrally symmetric convex ball $K$ (actually being the unit ball of a Minkowski normed space) with respect to a direction ${\bf x}$ of the Euclidean n-space $R^n$. We introduce the concept of general parameter spheres of $K$ corresponding to this direction and prove t…

We prove that the number of combinatorially distinct causal 3-dimensional triangulations homeomorphic to the 3-dimensional sphere is bounded by an exponential function of the number of tetrahedra. It is also proven that the number of combinatorially distinct causal 4-dimensional triangulations homeomorphic to the 4-sph…

Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.

We prove that the ending lamination space of the five-punctured sphere is homeomorphic to the Noebeling curve.

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

We prove that for any orientable connected surface of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, any homeomorphism of the space of geodesic laminations of this surface, equipped with the Thurston topology, is induced by a homeomorphism of the surface.

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via immersions if the map has no singular points on $M$ in the sense of F.H. Clarke, where $\dim M \leq \dim N$. As its corollary, we have that if a bi-Lipschitz homeomo…

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured sphere together with a map to the integers.…

3-manifolds are chiral if not finitely covered by sphere or product.

We consider when automorphisms of a graph can be induced by homeomorphisms of embeddings of the graph in a $3$-manifold. In particular, we prove that every automorphism of a graph is induced by a homeomorphism of some embedding of the graph in a connected sum of one or more copies of $S^2\times S^1$, yet there exist au…

We show that a free period three action on the three-sphere is standard, i.e. the quotient is homeomorphic to a lens space. We use a minimax argument involving sweepouts.

The paper classifies smooth structures on product manifolds of 3-connected 8-manifolds with spheres.

In this paper, we present a simple proof of the fact that any compact subgroup of homeomorphisms of the 2-sphere is topologically conjugate to a closed subgroup of the orthogonal group O(3).

In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…

We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.

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

The paper constructs contractible manifolds with knotted spheres.

The study of periodic subgroups in homeomorphism groups of manifolds.

We construct smooth 4-manifolds homeomorphic but not diffeomorphic to $CP^2#k\bar{CP^2},k \in {6,7,8,9}$, using the technique of rational blow-down along Wahl type plumbing trees of spheres.

Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.

Study rules out exotic $S^4$ and $\# n \mathbb{CP}^2$ construction using zero surgery homeomorphisms.

Hyperbolic groups act on spheres, and nearby actions are semi-conjugate.

Let G be a torsion-free hyperbolic group and let n > 5 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n-1)-dimensional sphere.

We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincaré homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-…

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such complete affine structures when Sigma is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Sigma, the deformation space identifies with…

The study examines how gamma positivity and PL homeomorphism types affect simplicial spheres.

We investigate a known problem whether a Sobolev homeomorphism between domains in $\mathbb{R}^n$ can change sign of the Jacobian. The only case that remains open is when $f\in W^{1,[n/2]}$, $n\geq 4$. We prove that if $n\geq 4$, and a sense-preserving homeomorphism $f$ satisfies $f\in W^{1,[n/2]}$, $f^{-1}\in W^{1,n-[n…

Let ${\mathcal H}$ be the group of isotopy classes of orientation preserving homeomorphisms of $S^3$ that preserve a Heegaard splitting of genus two. In this paper, we use a tree in the barycentric subdivision of the disk complex of a handlebody of the splitting to obtain a finite presentation of ${\mathcal H}$.

Classifies surfaces with great and small circles through each point.