arXiv research

Study noncommutative coverings of irrational quantum tori.

Simply-connected 4-manifolds are covered by products with a 2-torus.

We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spa…

Simply-connected surfaces of general type for n≥5.

We construct branched double coverings by certain direct products of manifolds for connected sums of copies of sphere bundles over the 2-sphere. As an application we answer a question of Kotschick and Loeh up to dimension five. More precisely, we show that: (1) every simply connected, closed four-manifold admits a bran…

Simply-connected shrinking Kähler-Ricci solitons are proven.

Given a closed oriented PL four-manifold $X$ and a closed surface $B$ embedded in $X$ with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of $X$ branched along $B$. For $X$ simply-connected, we deduce a necessary condition on the intersection form of a simply-connected i…

Flat connections induced over covering maps are studied and the trivial ones among them are described. In the sequel, we deal with the resulting holonomy bundles.

Research shows RCD* spaces are semi-locally simply connected.

Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space when developing their theory of generalized uniform co…

In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.

For a connected, locally path connected space $X$, let $H$ be a subgroup of the fundamental group of $X$, $π_1(X,x)$. We show that there exists an open cover $\cal U$ of $X$ such that $H$ contains the Spanier group $π({\U},x)$ if and only if the core of $H$ in $π_1(X,x)$ is open in the quasitopological fundamental grou…

Study on spectral stability of Riemannian coverings.

Functorial correspondence found between logarithmic connections and abelian connections.

Simple branched coverings established between certain 4-manifolds.

New knot invariants from covering involutions help deduce linear independence results.

The study connects lamination and orbit closures in hyperbolic manifolds.

This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connecte…

An oriented connected closed manifold $M^n$ is called a URC-manifold if for any oriented connected closed manifold $N^n$ of the same dimension there exists a nonzero degree mapping of a finite-fold covering $\widehat{M}^n$ of $M^n$ onto $N^n$. This condition is equivalent to the following: For any $n$-dimensional integ…

Given a smooth, oriented, closed surface $Σ$ of genus zero, possibly with boundary, let $\tildeΣ \longrightarrow Σ$ be a given $G$-cover of $Σ$, where $G$ is a given finite group. Let $S_{n}$ denote the standard sphere with $n$ holes. There are many ways of gluing together several $G$-cover of $S_{n}$ to construct the …

Compact manifolds with specific cover properties are hyperbolic.

Classifies 3-manifolds with distinct covers.

New method finds unrealizable branched cover data.

This paper classifies fundamental groups of perforated surfaces.

Conditions for simple closed curves in surface covers.

From the uniformization theorem, we know that every Riemann surface has a simply-connected covering space. Moreover, there are only three simply-connected Riemann surfaces: the sphere, the Euclidean plane, and the hyperbolic plane. In this paper, we collect the known heat kernels, or Green's functions, for these three …

Study shows no new eigenvalues in specific finite coverings.

Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.

Given a branched covering of degree d between closed surfaces, it determines a collection of partitions of d, the branch data. In this work we show that any branch data are realized by an indecomposable primitive branched covering on a connected close surface N with Euler's characteristic less than or equal to 0. This …

Paper derives Riccati equation for static spaces and proves its applications.

Study involutions on 3D small covers, proving quotient spaces are linked 2-spheres.

Infinite family of hyperbolic 3-manifolds with large volumes.

In this paper we study the connections between cyclic presentations of groups and branched cyclic coverings of (1,1)-knots. In particular, we prove that every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic presentation for the fundamental group encoded by a Heegaard diagram of genus n.

Open manifolds can be covered by $\mathbb{R}^m$ with finite or infinite degree.

Flat connections on bundles from noncommutative representations.

Support theorem proved for X-ray transform on certain non-compact manifolds.

The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; an…

Enhances understanding of stability conditions on surfaces.

We obtain several results about stability of the Bergman kernel on a tower of coverings on complex manifolds. An effective version of Rhodes' result is given for a tower of coverings on a compact Riemann surface of genus greater than or equal to 2. Stability of the Bergman kernel is established for towers of coverings …

Integral foliated simplicial volume is zero for certain amenable covers.

The paper classifies almost complex manifolds with non-degenerate torsion and their coverings.

We show that the commutator subgroup G' of a classical knot group G need not have subgroups of every finite index, but it will if G' has a surjective homomorphism to the integers and we give an exact criterion for that to happen. We also give an example of a smoothly knotted n-sphere in the (n+2)-sphere for all n at le…

Study bounds Urysohn width of manifolds under surgeries.

Study fundamental groups of small covers and their injective submanifolds.

Finite vector bundles over complex manifolds are trivializable via finite covers.

The study of Platonic solids' unfoldings leads to high genus Teichmüller curves.

Let N be a regular branched cover of a homology 3-sphere M with deck group G isomorphic to Z_2^d and branch set a trivalent graph Gamma; such a cover is determined by a coloring of the edges of Gamma with elements of G. For each index-2 subgroup H of G, M_H = N/H is a double branched cover of M. Sakuma has proved that …

Computes constants for cyclic covers of translation surfaces.