Genus sets and SNT sets of certain connective covering spaces

Huale Huang; Joseph Roitberg

Displaying similar documents to “Genus sets and SNT sets of certain connective covering spaces”

Involutions of 3-dimensional handlebodies

Andrea Pantaleoni, Riccardo Piergallini (2011)

Fundamenta Mathematicae

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We study the orientation preserving involutions of the orientable 3-dimensional handlebody $H_{g}$ , for any genus g. A complete classification of such involutions is given in terms of their fixed points.

Lifting of homeomorphisms to branched coverings of a disk

Bronisław Wajnryb, Agnieszka Wiśniowska-Wajnryb (2012)

Fundamenta Mathematicae

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We consider a simple, possibly disconnected, d-sheeted branched covering π of a closed 2-dimensional disk D by a surface X. The isotopy classes of homeomorphisms of D which are pointwise fixed on the boundary of D and permute the branch values, form the braid group Bₙ, where n is the number of branch values. Some of these homeomorphisms can be lifted to homeomorphisms of X which fix pointwise the fiber over the base point. They form a subgroup $L^{π}$ of finite index in Bₙ. For each equivalence...

An upper bound for the minimum genus of a curve without points of small degree

Claudio Stirpe (2013)

Acta Arithmetica

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We prove that for any prime p there is a constant Cₚ > 0 such that for any n > 0 and for any p-power q there is a smooth, projective, absolutely irreducible curve over $_{q}$ of genus g ≤ Cₚqⁿ without points of degree smaller than n.

Computing the determinantal representations of hyperbolic forms

Mao-Ting Chien, Hiroshi Nakazato (2016)

Czechoslovak Mathematical Journal

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The numerical range of an $n \times n$ matrix is determined by an $n$ degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an $n$ degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus $g = 1$ . We reformulate the Fiedler-Helton-Vinnikov formulae for the genus $g = 0, 1$ , and present an elementary...

Quandle coverings and their Galois correspondence

Michael Eisermann (2014)

Fundamenta Mathematicae

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This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p: (Q̃,q̃̃) → (Q,q). This in turn leads us to define the algebraic fundamental group $π ₁ (Q, q) : = A u t (p) = g \in A d j {(Q)}^{'} | q^{g} = q$ , where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected coverings of (Q,q) and subgroups of π₁(Q,q). Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble...

The covering number for category and partition relations on $P_{ω} (λ)$

Pierre Matet (2002)

Fundamenta Mathematicae

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We show that cov(M) is the least infinite cardinal λ such that $P_{ω} (λ)$ (the set of all finite subsets of λ ) fails to satisfy a certain natural generalization of Ramsey’s Theorem.

Dimension vs. genus: A surface realization of the little k-cubes and an $E_{\infty}$ operad

Ralph M. Kaufmann (2009)

Banach Center Publications

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We define a new $E_{\infty}$ operad based on surfaces with foliations which contains $E_{k}$ suboperads. We construct CW models for these operads and provide applications of these models by giving actions on Hochschild complexes (thus making contact with string topology), by giving explicit cell representatives for the Dyer-Lashof-Cohen operations for the 2-cubes and by constructing new Ω spectra. The underlying novel principle is that we can trade genus in the surface representation vs. the dimension...

Irreducibility of Hurwitz Spaces of Coverings with Monodromy Groups Weyl Groups of Type $W(B_{d})$

Francesca Vetro (2007)

Bollettino dell'Unione Matematica Italiana

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Let Y be a smooth, connected, projective complex curve of genus > = 0. Biggers and Fried proved the irreducibility of the Hurwitz spaces which parametrize coverings of $P^{1}$ whose monodromy group is a Weyl of type $W(B_{d})$ . Here we prove the irreducibility of Hurwitz spaces that parametrize coverings of Y with monodromy group a Weyl group of type $W(B_{d})$ .

On the arithmetic of the hyperelliptic curve $y^{2} = x^{n} + a$

Kevser Aktaş, Hasan Şenay (2016)

Czechoslovak Mathematical Journal

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We study the arithmetic properties of hyperelliptic curves given by the affine equation $y^{2} = x^{n} + a$ by exploiting the structure of the automorphism groups. We show that these curves satisfy Lang’s conjecture about the covering radius (for some special covering maps).

Non-standard automorphisms of branched coverings of a disk and a sphere

Bronisław Wajnryb, Agnieszka Wiśniowska-Wajnryb (2012)

Fundamenta Mathematicae

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Let Y be a closed 2-dimensional disk or a 2-sphere. We consider a simple, d-sheeted branched covering π: X → Y. We fix a base point A₀ in Y (A₀ ∈ ∂Y if Y is a disk). We consider the homeomorphisms h of Y which fix ∂Y pointwise and lift to homeomorphisms ϕ of X-the automorphisms of π. We prove that if Y is a sphere then every such ϕ is isotopic by a fiber-preserving isotopy to an automorphism which fixes the fiber $π^{- 1} (A ₀)$ pointwise. If Y is a disk, we describe explicitly a small set of automorphisms...

Erratum to the paper "On the disc theorem" (Ann. Polon. Math. 55 (1991), 1-10)

Cabiria Andreian Cazacu (1992)

Annales Polonici Mathematici

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Due to a technical error, part of a sentence was omitted on the top of page 8. The first line should read: “where $f_{p}^{k}$ , $p = a_{l}$ or $b_{l}$ , means the number of folds of the covering $(δ_{k}^{''}, T |, Δ_{l}^{''})$ ending at p, i.e. covering a neighbourhood of p in $a_{l} b_{l}$ without covering p itself”.

A new lower bound for the football pool problem for $7$ matches

Laurent Habsieger (1996)

Journal de théorie des nombres de Bordeaux

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Let $K_{3} (7, 1)$ denote the minimum cardinality of a ternary code of length $7$ and covering radius one. In a previous paper, we improved on the lower bound $K_{3} (7, 1) \geq 147$ by showing that $K_{3} (7, 1) \geq 150$ . In this note, we prove that $K_{3} (7, 1) \geq 153$ .

On Ponomarev-Systems

Ying Ge, Lin Shou (2007)

Bollettino dell'Unione Matematica Italiana

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In this paper the relations of mappings and families of subsets are investigated in Ponomarev-systems, and the following results are obtained. (1) $f$ is a sequence-covering (resp. 1-sequence-covering) mapping iff $\mathcal{P}$ is a csf -network (resp. snf -network) of $X$ for a Ponomarev-system $(f,M,X,\mathcal{P})$ ; (2) $f$ is a sequence-covering (resp. 1-sequence-covering) mapping iff every $\mathcal{P}_{n}$ is a cs-cover (resp. wsn-cover) of $X$ for a Ponomarev-system $(f,M,X,\{\mathcal{P}_{n}\})$ . As applications of these results, some relations between sequence-covering...

Some topological properties of $ω$ -covering sets

Andrzej Nowik (2000)

Czechoslovak Mathematical Journal

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We prove the following theorems: There exists an $ω$ -covering with the property $s_{0}$ . Under $c o v (𝒩) =$ there exists $X$ such that $\forall_{B \in ℬ o r} [B \cap X$ is not an $ω$ -covering or $X ∖ B$ is not an $ω$ -covering]. Also we characterize the property of being an $ω$ -covering.

On signatures associated with ramified coverings and embedding problems

J. Wood, Emery Thomas (1973)

Annales de l'institut Fourier

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Given a cohomology class $ξ \in H^{2} (M; Z)$ there is a smooth submanifold $K \subset M$ Poincaré dual to $ξ$ . A special class of such embeddings is characterized by topological properties which hold for nonsingular algebraic hypersurfaces in $C P_{n}$ . This note summarizes some results on the question: how does the divisibility of $ξ$ restrict the dual submanifolds $K$ in this class ? A formula for signatures associated with a $d$ -fold ramified cover of $M$ branched along $K$ is given and a proof is included in case $d = 2$ .

The genera, reflexibility and simplicity of regular maps

Marston Conder, Jozef Širáň, Thomas Tucker (2010)

Journal of the European Mathematical Society

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This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to $g - 1$ , where $g$ is the genus, all orientably-regular maps of genus $p + 1$ for $p$ prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and...

A proof of the crossing number of $K_{3, n}$ in a surface

Pak Tung Ho (2007)

Discussiones Mathematicae Graph Theory

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In this note we give a simple proof of a result of Richter and Siran by basic counting method, which says that the crossing number of $K_{3, n}$ in a surface with Euler genus ε is ⎣n/(2ε+2)⎦ n - (ε+1)(1+⎣n/(2ε+2)⎦).

Green's generic syzygy conjecture for curves of even genus lying on a K3 surface

Claire Voisin (2002)

Journal of the European Mathematical Society

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We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: For a smooth projective curve $C$ of genus $g$ in characteristic 0, the condition $Cliff C > l$ is equivalent to the fact that $K_{g - l^{'} - 2, 1} (C, K_{C}) = 0, \forall l^{'} \leq l$ . We propose a new approach, which allows up to prove this result for generic curves $C$ of genus $g (C)$ and gonality $gon(C)$ in the range $\frac{g (C)}{3} + 1 \leq gon(C) \leq \frac{g (C)}{2} + 1 .$

On covering and quasi-unsplit families of curves

Laurent Bonavero, Cinzia Casagrande, Stéphane Druel (2007)

Journal of the European Mathematical Society

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Given a covering family $V$ of effective 1-cycles on a complex projective variety $X$ , we find conditions allowing one to construct a geometric quotient $q : X \to Y$ , with $q$ regular on the whole of $X$ , such that every fiber of $q$ is an equivalence class for the equivalence relation naturally defined by $V$ . Among other results, we show that on a normal and $ℚ$ -factorial projective variety $X$ with canonical singularities and $dim X \leq 4$ , every covering and quasi-unsplit family $V$ of rational curves generates a geometric...