Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0709.0150

Realizability and exceptionality of candidate surface branched covers: methods and results

Ekaterina Pervova·Carlo Petronio
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 has positive genus. For the sphere one knows that for some data the cover exists and for some it does not, but the problem is still open in general. In this paper we will review five different techniques recently employed to attack it, and we will state the main results they have led to. To illustrate the techniques we will give five independent proofs of the fact that there is no branched cover of the sphere over itself with degree 4, three branching points, and local degrees (2,2), (2,2), and (3,1) over them (despite the fact that the Riemann-Hurwitz formula is satisfied).

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ekaterina Pervova, Carlo Petronio. 2007-09-10. Realizability and exceptionality of candidate surface branched covers: methods and results. https://arxiv.org/abs/0709.0150

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

3d Super Hyperbolic Geometry

Wigner's unitary representation of the Lorentz group is extended to a representation of the complex orthosymplectic Lie supergroup (\operatorname{OSp}_{\mathbb C}(1|2)) acting on Minkowski ((3,1|4))-dimensional superspace essentially by Hermitian conjugation. The invariant quadratic form is[x_1x_2-x\bar x+ϕψ+\barϕ\barψ]in Wigner's real coordinates (x_1,x_2) and complex coordinate (x), where (ϕ) and (ψ) are Dirac fermions. The extended action is linear in the superspace variables, but not quadratic in the odd group variables, and is described by an explicit even purely imaginary auxiliary parameter defined on the product of the Lie supergroup with its Lie algebra. This extension of Wigner's representation opens the door to the study of geometry in super hyperbolic three-space, initiated here with foundations on super geodesics, triangles, and tetrahedra, together with an analysis of ideal tetrahedra and a discussion of the nonzero fermionic correction to the Schläfli formula. This version includes a corrigendum that does not materially affect the main results, except for the claimed divergence of the volume of an ideal tetrahedron: the earlier divergence computation was correct as a calculation, but was carried out for an incorrect differential form.

math.GT↗
arXiv2026

Seifert cobordisms and the Chen-Yang volume conjecture

We study the large $r$ asymptotic behavior of the Turaev-Viro invariants $TV_r(M; e^{\frac{2πi}{r}})$ of 3-manifolds with toroidal boundary, under the operation of gluing a Seifert-fibered 3-manifold along a component of $\partial M$. We show that the Turaev-Viro invariants volume conjecture is closed under this operation. As an application we prove the volume conjecture for all Seifert fibered 3-manifolds with boundary and for large classes of graph 3-manifolds.

math.GT↗
arXiv2026

Equivariant lattice homology for Seifert homology 3-spheres

We develop an $S^1 \times \mathbb{Z}_p$-equivariant refinement of the lattice homotopy type for Seifert homology 3-spheres equipped with the Seifert $\mathbb{Z}_p$-action. We construct an equivariant comparison map to the $S^1 \times \mathbb{Z}_p$-equivariant Seiberg--Witten Floer spectrum that becomes an $S^1$-equivariant stable homotopy equivalence after forgetting the $\mathbb{Z}_p$-action, yielding a combinatorial description of its Borel equivariant cohomology.

math.GT↗