arXiv research

Generalizes Leighton's theorem to cube complexes.

Two complexes share a common covering but not a finite one.

Tiny complexes share 3-5 triangles in common coverings.

A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are …

Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.

Minimal example found for two finite CW-complexes sharing a common covering.

Paper proves ratios of Chern numbers differ for complex hyperbolic branched covers.

The paper presents a chain complex for 3-manifold covers, including surface bundles and surgeries.

Study connects manifold complexity to scalar curvature bounds.

Corks transform complex curves without changing topology.

New complexity measure ADL connects to classical complexity measures.

The notion of covering type was recently introduced by Karoubi and Weibel to measure the complexity of a topological space by means of good coverings. When X has the homotopy type of a finite CW-complex, its covering type coincides with the minimum possible number of vertices of a simplicial complex homotopy equivalent…

New criterion for branched covers between 2-spheres.

A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem …

The paper classifies sextic curves on a Fano 3-fold with rational Galois covers in 3D space.

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

We study branched covering spaces in several contexts, proving that under suitable circumstances the cover satisfies the same upper curvature bounds as the base space. The first context is of a branched cover of an arbitrary metric space that satisfies Alexandrov's curvature condition CAT(k), over an arbitrary complete…

This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.

Characterizes compact complex surfaces with finite homotopy rank-sum.

In "Rips complexes and covers in the uniform category" \cite{Rips} the authors define, following James \cite{J}, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given…

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

In this paper, we study the properties of coverings of locally conformally Kähler (LCK) spaces with singularities. We begin by proving that a space is LCK if any only if its universal cover is Kähler, thereby generalizing a result from a previous paper. We then show that a complex space which projects over an LCK space…

We provide the first non-trivial examples of quasi-isometric embeddings between curve complexes. These are induced either by puncturing a closed surface or via orbifold coverings. As a corollary, we give new quasi-isometric embeddings between mapping class groups.

Characterizes covers using simple closed curves on surfaces.

We show a Whitney Approximation Theorem for a continuous map from a manifold to a smooth CW complex. This enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It is also shown that, for any open covering of a smooth CW complex, there exists …

Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.

A closed 4-manifold (or, more generally, a finite $PD_4$-space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus of a self homotopy equivalence of a $PD_3$-complex.

We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that…

Study properties of balanced hyperbolic compact complex manifolds.

Croke and Kleiner constructed two homeomorphic locally CAT(0) complexes whose universal covers have visual boundaries that are not homeomorphic. We construct two homeomorphic locally CAT(0) complexes so that the visual boundary of one universal cover contains a nonplanar graph, while the visual boundary of the other do…

Proves properties of complex algebraic varieties and local systems.

\noindent Given a Riemann surface $M$, the \emph{complexity} of a branched cover of $M$ to the Riemann sphere $S^2$, of degree $d$ and with branching set of cardinality $n \geq 3$, is defined as $d$ times the hyperbolic area of the complement of its branching set in $S^2$. A branched cover $p \colon M \to S^2$ of degre…

Rafi and Schleimer recently proved that the natural relation between curve complexes induced by a covering map between two surfaces is a quasi-isometric embedding. We offer another proof of this result using a distance estimate via hyperbolic 3-manifolds.

In this paper we develop analysis of the monopole maps over the universal covering space of a compact four manifold. We induce a property on local properness of the covering monopole map under the condition of closeness of the AHS complex. In particular we construct a higher degree of the covering monopole map when the…

We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X. In the process, we identify the d^1 differential in terms of the coalgebra structure of H_*(X,\k), and the \kπ_1(X)-module struct…

We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the covering for the Matveev complexity of the covering manifold. The lower estimate …

We construct a generalization of twistor spaces of hypercomplex manifolds and hyper-Kahler manifolds $M$, by generalizing the twistor $\mathbb{P}^{1}$ to a more general complex manifold $Q$. The resulting manifold $X$ is complex if and only if $Q$ admits a holomorphic map to $\mathbb{P}^1$. We make branched double cove…

We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of ran…

New complex connects graph separability to group properties.

This paper extends FP's method to complex hyperbolic branched covers to find Einstein metrics.

Isometric embeddings of Teichmüller spaces are derived from branched coverings.

Counterexample found for Stein property of certain solvable Lie groups.

Study 2-complexes' homology properties and torsion growth.

We prove an effective version of a theorem relating curve complex distance to electric distance in hyperbolic 3-manifolds, up to errors that are polynomial in the complexity of the underlying surface. We use this to give an effective proof of a result regarding maps between curve complexes of surfaces induced by finite…

We show that if a bounded domain in complex Euclidean space with $\mathcal{C}^{1,1}$ boundary covers a compact manifold, then the domain is biholomorphic to the unit ball.

We study the de Rham complex on a smooth manifold with a periodic end modeled on an infinite cyclic cover X' \to X. The completion of this complex in exponentially weighted L^2-norms is Fredholm for all but finitely many exceptional weights determined by the eigenvalues of the covering translation map H_*(X') \to H_*(X…

New bounds for private learning of high-dimensional Gaussian distributions.

The study finds a limit on subgroup complexity in hyperbolic 3-manifold groups.