arXiv research

For surface-knots, branched covers with degree 3 have simplifying numbers <3.

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

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

For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whe…

Lower bounds for cover degrees of hyperbolic 3-manifolds.

To a branched cover between closed, connected and orientable surfaces one associates a "branch datum", which consists of the two surfaces, the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann-Hurwitz formula. A "candidate su…

In this work we characterize branch data of branched coverings of even degree over the projective plane which are realizable by indecomposable branched coverings.

New covering moves for 3-manifolds up to degree 4.

Researchers study chirality in a specific type of torus-covering link.

The paper studies random covers of torus knot complements and their statistical properties.

The paper finds new graph covers with exceptionally low degree.

3-manifolds can be virtually dominated by maps of degree 8.

Study shows indecomposable branched coverings over projective plane for surfaces with low Euler characteristic.

Given two closed orientable surfaces, the Hurwitz existence problem asks whether there exists a branched cover between them having prescribed global degree and local degrees over the branching points. The Riemann-Hurwitz formula gives a necessary condition, which was shown to be also sufficient when the base surface ha…

Computes constants for cyclic covers of translation surfaces.

For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary re…

We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S^3. We also prove a stabilization result: after adding a fifth trivial sheet two local moves suffice. These results are analogous to results of Piergallini in degr…

New partial solution to Hurwitz problem for surface branched covers.

Gromov-Thurston covers have Betti numbers as expected.

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…

New bounds on cover degrees for Teichmüller distance between hyperbolic surfaces.

Researchers compute specific types of branched surface covers.

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…

New proof shows surfaces can have identical length spectra but not simple ones.

Study various complexity measures of curves on surfaces.

Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavio…

We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over…

Classifies 3-manifolds with distinct covers.

3-manifolds can virtually dominate others with positive simplicial volume.

Loi and Piergallini showed that a smooth compact, connected $4$-manifold $X$ with boundary admits a Stein structure if and only if $X$ is a simple branched cover of a $4$-disk $D^4$ branched along a positive braided surface $S$ in a bidisk $D_{1}^{2} \times D_{2}^{2} \approx D^4$. For each integer $N \geq 2$, we constr…

Paper estimates area covered by a line-sweep sensor in robotics.

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 …

We study how the systole of principal congruence coverings of a Hilbert modular variety grows when the degree of the covering goes to infinity. We prove that given a Hilbert modular variety $M$ of real dimension $2n$, the sequence of principal congruence coverings $M_{I}$ eventually satisfies $$sysπ_{1}(M_{I})\geq \fra…

Study smooth linear statistics on random covers of hyperbolic surfaces, showing central limit and variance results.

We prove that while there are maps $\bT^4\to\#^3(\bS^2\times\bS^2)$ of arbitrarily large degree, there is no branched cover from $4$-torus to $\#^3(\bS^2\times \bS^2)$. More generally, we obtain that, as long as $N$ satisfies a suitable cohomological condition, any $π_1$-surjective branched cover $\bT^n \to N$ is a hom…

Random covers of surfaces have tangle-free monodromy and the Putman-Wieland property.

Researchers compute specific Hurwitz numbers for branched covers.

We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover away from $D$ has degree $d=1$ or $d=\infty$ (however all degrees $d > 1$ are poss…

K-stability proven for a specific type of Fano threefold.

Elementary proof shows no specific torus to sphere cover with certain branching points.

The paper constructs simplicial maps of any degree on spheres, solving a long-standing problem.

Stability in homology of moduli spaces of admissible covers.

\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…

Random covers of hyperbolic surfaces have a spectral gap with polynomial rate.

Let $M$ be a closed manifold that admits a self-cover $p:M \to M$ of degree >1. We say p is strongly regular if all its iterates are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of $M$: We prove that $π_1(M)$ surjects onto a nontrivial free abelian group $A$, and t…

A quick proof of Bing's theorem indicated by the title is given. The proof also concludes Gumerov's result on covering degrees of solenoids.

Special covers of alternating links have finite index subgroups in certain groups.

The paper characterizes isomorphic covers of surfaces and applies it to distinguish representations.