Cyclic Coverings Research Articles

Theorem on the union of two topologically flat cells of codimension 1 in ℝn

In this paper we give a complete and improved proof of the “Theorem on the union of two (n − 1)‐cells.” First time it was proved by the author in the form of reduction to the earlier author′s technique. Then the same reduction by the same method was carried out by Kirby. The proof presented here gives a more clear reduction. We also present here the exposition of this technique in application to the given task. Besides, we use a modification of the method, connected with cyclic ramified coverings, that allows us to bypass referring to the engulfing lemma as well as to other multidimensional results, and so the theorem is proved also for spaces of any dimension. Thus, our exposition is complete and does not require references to other works for the needed technique.

Classifying cubic symmetric graphs of order 10p or 10p 2

A graph is called s -regular if its automorphism group acts regularly on the set of its s -arcs. In this paper, the s -regular cyclic or elementary abelian coverings of the Petersen graph for each s ≥1 are classified when the fibre-preserving automorphism groups act arc-transitively. As an application of these results, all s -regular cubic graphs of order 10 p or 10 p 2 are also classified for each s ≥1 and each prime p , of which the proof depends on the classification of finite simple groups.

Asymptotic windings over the trefoil knot

Consider the group G:=PSL_2(\mathbb R) and its subgroups \Gamma:= PSL_2(\mathbb{Z}) and \Gamma':= DSL_2(\mathbb{Z}) . G/\Gamma is a canonical realization (up to an homeomorphism) of the complement \mathbb S^3\setminus T of the trefoil knot T , and G/\Gamma' is a canonical realization of the 6-fold branched cyclic cover of \mathbb S^3\setminus T , which has 3-dimensional cohomology of 1-forms. Putting natural left-invariant Riemannian metrics on G , it makes sense to ask which is the asymptotic homology performed by the Brownian motion in G/\Gamma' , describing thereby in an intrinsic way part of the asymptotic Brownian behavior in the fundamental group of the complement of the trefoil knot. A good basis of the cohomology of G/\Gamma' , made of harmonic 1-forms, is calculated, and then the asymptotic Brownian behavior is obtained, by means of the joint asymptotic law of the integrals of the above basis along the Brownian paths. Finally the geodesics of G are determined, a natural class of ergodic measures for the geodesic flow is exhibited, and the asymptotic geodesic behavior in G/\Gamma' is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).

Cyclic Difference Packing and Covering Arrays

Let n and k be positive integers. Let Cq be a cyclic group of order q. A cyclic difference packing (covering) array, or a CDPA(k, n; q) (CDCA(k, n; q)), is a k × n array (aij) with entries aij (0 ? i ? k?1, 0 ? j ? n?1) from Cq such that, for any two rows t and h (0 ? t < h ? k?1), every element of Cq occurs in the difference list $${\Delta}_{th} = {d_{hj}- d_{tj}: j = 0, 1, \dots, n-1}$$ at most (at least) once. When q is even, then n ? q?1 if a CDPA(k, n; q) with k ? 3 exists, and n ? q+1 if a CDCA(k, n; q) with k ? 3 exists. It is proved that a CDCA(4, q+1; q) exists for any even positive integers, and so does a CDPA(4, q?1; q) or a CDPA(4, q?2; q). The result is established, for the most part, by means of a result on cyclic difference matrices with one hole, which is of interest in its own right.

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi–Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer–Severi varieties and, in the quaternion case, we show how to obtain some Brauer–Severi varieties from G-Hilbert schemes of P 1 -bundles.

ON CERTAIN CLASSES OF LINKS AND 3-MANIFOLDS

We construct an infinite family of closed 3-manifolds M(2m+ 1, n, k) which are identification spaces of certain polyhedra P(2m+ 1, n, k), for integers <TEX>$m\;\ge\;1,\;n\;\ge\;3,\;and\;k\;\ge\;2$</TEX>. We prove that they are (n / d)- fold cyclic coverings of the 3-sphere branched over certain links <TEX>$L_{(m,d)}$</TEX>, where d = gcd(n, k), by handle decomposition of orbifolds. This generalizes the results in [3] and [2] as a particular case m = 2.

Automorphism groups for p-cyclic covers of the affine line

let k be an algebraically closed field of positive characteristic p > 0 and $c \to {\mathbb p}^1_k$ a p -cyclic cover of the projective line ramified in exactly one point. we are interested in the p -sylow subgroups of the full automorphism group aut k c . we prove that for curves c with genus 2 or higher, these groups are exactly the extensions of a p -cyclic group by an elementary abelian p -group. the main tool is an efficient algorithm to compute the p -sylow subgroups of aut k c starting from an artin–schreier equation for the cover $c \to {\mathbb p}^1_k$ . we also characterize curves c with genus $g_c\geq 2$ and a p -group action $g\subset \text{aut}_k c$ such that $2p/(p-1) and $4/(p-1)^2\leq |g|/g_c^2$ . our methods rely on previous work by stichtenoth whose approach we have adopted.

On symplectic coverings of the projective plane

We prove that a resolution of singularities of any finite covering of the projective complex plane branched along a Hurwitz curve , and possibly along the line “at infinity”, can be embedded as a symplectic submanifold in some projective algebraic manifold equipped with an integer Kähler symplectic form. (If has negative nodes, then the covering is assumed to be non-singular over them.) For cyclic coverings, we can realize these embeddings in a rational complex 3-fold. Properties of the Alexander polynomial of are investigated and applied to the calculation of the first Betti number , where is a resolution of singularities of an -sheeted cyclic covering of branched along , and possibly along the line “at infinity”. We prove that is even if is an irreducible Hurwitz curve but, in contrast to the algebraic case, may take any non-negative value in the case when consists of several components.

An infinite family of non-concordant knots having the same Seifert form

By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this paper, we extend this result to the full extent. We show that if the knot has nontrivial Alexander polynomial, then there exists an infinite family of non-concordant knots having the same Seifert form as the knot. As a corollary, no nontrivial Alexander polynomial determines a unique knot concordance class. We use Cochran-Orr-Teichner's recent result on the knot concordance group and Cheeger-Gromov's von Neumann \rho -invariants with their universal bound for a 3-manifold.

Vanishing thetanulls for some dihedral and cyclic coverings of Riemann surfaces

Let Wg → Wz be a ramified p-sheeted covering of Riemann surfaces of genus g and z, (z > 0) where p is an odd prime. Assume that the Galois group is either dihedral or cyclic. Assume, moreover, that the covering is full; that is, there us an integral divisor E, of degree 2r on Wz which lifts to be canonical on Wg. Then g = rp + 1, where r ≥ 1. Clearly, Wg admits 22z half-canonical linear series of dimension at least r − z arising from divisors on Wz whose double is E. Theorem 1 Of these 22z half-canonical linear series uz (= 2z−1 (2z−1)) have dimension at least r − z + 1. Theorem 2 Let Wg (g = 3r + 1, r ≥ 3) admit four half canonical linear series, three of dimension r − 1, and one of dimension r, whose sum is bi-canonical, where the half-canonical linear series of dimension r is unique. Then Wg is a full elliptic-trigonal Riemann surface. (This characterizes the cases z = 1, p = 3, g ≥ 10)

Theta constants associated to coverings of P1 branching at eight points

In this paper, we construct automorphic forms on the five-dimensional complex ball which give the inverse of the period map for cyclic 4-ple coverings of the complex projective line branching at eight points. We use theta constants associated to the Prym varieties of these coverings.

Filtration of the classical knot concordance group and Casson–Gordon invariants

It is known that if any prime power branched cyclic cover of a knot in the 3-sphere is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in the 3-sphere whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of F_(1.0)/F_(1.5) for which Casson-Gordon invariants vanish in Cochran-Orr-Teichner's filtration of the classical knot concordance group . As a corollary, it follows that Casson-Gordon invariants are not a complete set of obstructions to a second layer of Whitney disks.

Equations of low-degree projective surfaces with three-divisible sets of cusps

Let Y be a surface with only finitely many singularities all of which are cusps. A set of cusps on Y is called three-divisible, if there is a cyclic global triple cover of Y branched precisely over these cusps. The aim of this note is to determine the equations of surfaces $Y \subset P_3$ of degrees $\leq 6$ carrying a minimal, non-empty, three-divisible set.

ON INFINITE CLASSES OF GENUS TWO 1-BRIDGE KNOTS

We study a family of 2-bridge knots with 2-tangles in the 3-sphere admitting a genus two 1-bridge splitting. We also observe a geometric relation between (g i 1;1)-splitting and (g;0)- splitting for g = 2;3. Moreover we construct a family of closed orientable 3-manifolds which are n-fold cyclic coverings of the 3- sphere branched over those 2-bridge knots. 1. Preliminaries and deflnitions By a handlebody, we mean a bounded connected oriented 3-manifold V which admits mutually disjoint proper disks such that they split V into solid tori. Let M be a closed connected oriented 3-manifold. A Heegaard handlebody of M is a handlebody V in M such that V 0 = Cl(M i V ), the closure of M i V , is also a handlebody. The surface H = V V 0 and V (H V 0 are called a Heegaard surface(or Heegaard diagram) and a Heegaard splitting of M, respectively. Every closed connected oriented 3-manifold M admits a Heegaard splitting.

The stack of Potts curves and its fibre at a prime of wild ramification

In this note we study the modular properties of a family of cyclic coverings of P 1 of degree N, in all odd characteristics. We compute the moduli space of the corresponding algebraic stack over Z[1/2] , as well as the Picard groups over algebraically closed fields. We put special emphasis on the study of the fibre of the stack at a prime of wild ramification; in particular we show that the moduli space has good reduction at such a prime.

Linking numbers in rational homology 3-spheres, cyclic branched covers and infinite cyclic covers

We study the linking numbers in a rational homology 3 3 -sphere and in the infinite cyclic cover of the complement of a knot. They take values in Q \mathbb {Q} and in Q ( Z [ t , t − 1 ] ) {Q}(\mathbb {Z}[t,t^{-1}]) , respectively, where Q ( Z [ t , t − 1 ] ) {Q}(\mathbb {Z}[t,t^{-1}]) denotes the quotient field of Z [ t , t − 1 ] \mathbb {Z}[t,t^{-1}] . It is known that the modulo- Z \mathbb {Z} linking number in the rational homology 3 3 -sphere is determined by the linking matrix of the framed link and that the modulo- Z [ t , t − 1 ] \mathbb {Z}[t,t^{-1}] linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate ‘modulo Z \mathbb {Z} ’ and ‘modulo Z [ t , t − 1 ] \mathbb {Z}[t,t^{-1}] ’. When the finite cyclic cover of the 3 3 -sphere branched over a knot is a rational homology 3 3 -sphere, the linking number of a pair in the preimage of a link in the 3 3 -sphere is determined by the Goeritz/Seifert matrix of the knot.

S-REGULAR DIHEDRAL COVERINGS OF THE COMPLETE GRAPH OF ORDER 4

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. An infinite family of cubic 1-regular graphs was constructed in [7] as cyclic coverings of the three-dimensional Hypercube, and a classification of all s-regular cyclic coverings of the complete bipartite graph of order 6 was given in [8] for each s≥1, whose fibre-preserving automorphism subgroups act arc-transitively. In this paper, the authors classify all s-regular dihedral coverings of the complete graph of order 4 for each s≥1, whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed.

S‐Regular cubic graphs as coverings of the complete bipartite graph K3,3

AbstractA graph is s‐regular if its automorphism group acts freely and transitively on the set of s‐arcs. An infinite family of cubic 1‐regular graphs was constructed in [10], as cyclic coverings of the three‐dimensional Hypercube. In this paper, we classify the s‐regular cyclic coverings of the complete bipartite graph K3,3 for each ≥ 1 whose fibre‐preserving automorphism subgroups act arc‐transitively. As a result, a new infinite family of cubic 1‐regular graphs is constructed. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 101–112, 2004

Regular cyclic coverings of regular affine maps

The regular coverings of regular affine algebraic maps are considered, and a large family of totally ramified coverings—the so-called Steinberg and Accola coverings—are fully classified.

S-Regular cyclic coverings of the three-dimensional hypercube Q3

A graph is s- regular if its automorphism group acts regularly on the set of s-arcs. An infinite family of cubic 1-regular graphs was constructed in European J. Combin. 23 (2002) 559, as cyclic coverings of the three-dimensional hypercube Q 3. In this paper, we classify the s-regular cyclic coverings of Q 3 for each s≥1, whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed.