Cyclic Coverings Research Articles

Ramification points of double coverings of curves and Weierstrass points

Let π: X→C be a double covering with X smooth curve and C elliptic curve. Let R(π)⊂X be the ramification locus of π. Every P∈R(π) is a Weierstrass point of X and we study the triples (C, π, X) for which the set of corresponding Weierstrass points have certain semigroups of non-gaps. We study the same problem also for triple cyclic coverings of C.

On the cyclic coverings of the knot 52

We construct a family of hyperbolic 3-manifolds whose fundamental groups admit a cyclic presentation. We prove that all these manifolds are cyclic branched coverings of S3 over the knot 52 and we compute their homology groups. Moreover, we show that thecyclic presentations correspond to spines of the manifolds.

Spinning and branched cyclic covers of knots

A necessary and sufficient algebraic condition is given for a Z–torsion–free simple 2 q –knot, q ⩾ 3, to be the r –fold branched cyclic cover of a knot. This result is then used to give a necessary and sufficient algebraic condition on a simple (2 q −1)–knot for its spun knot to be the r –fold branched cyclic cover of a knot. Finally, we give examples for q ⩾ 5 of a simple (2 q −1)–knot which is not the 2 r –fold branched cyclic cover of any knot although its spun knot is.

Topological Quantum Field Theory and Strong Shift Equivalence

AbstractGiven a TQFT in dimensiond+ 1; and an infinite cyclic covering of a closed (d+ 1)-dimensional manifoldM, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams’ work in symbolic dynamics. The Turaev-Viro module associated to a TQFT and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary ofMhas anS1factor and the infinite cyclic cover of the boundary is standard. We define a variant of a TQFT associated to a finite groupGwhich has been studied by Quinn. In this way, we recover a link invariant due to D. Silver and S. Williams. We also obtain a variation on the Silver-Williams invariant, by using the TQFT associated toGin its unmodified form.

Acyclic covers

The following question, which is directly related to the Whitehead problem of subcomplexes of acyclic 2-complexes, is studied: If $\frak P$ is a class of groups, X is a 2-dimensional CW-complex and X' is an acyclic, infinite cyclic cover of X with $\pi\_1(X')$ in $\frak P$ , must X' be contractible? A positive answer is given if X is finite and $\frak P$ is the class of amenable groups.

ON THE ASYMPTOTICS OF MORSE NUMBERS OF FINITE COVERS OF MANIFOLDS

Let M be a closed connected manifold. We denote by M(M) the Morse number of M, i.e. the minimal possible number of critical points of a Morse function f on M. M.Gromov posed the following question: Let N k, k∈ N be a sequence of manifolds, such that each N k is an a k -fold cover of M where a k →∞ as k→∞. What are the asymptotic properties of the sequence M(N k) as k→∞? In this paper we study the case π 1(M)≈ Z m, dim M⩾6 . Let ξ∈H 1(M, Z), ξ≠0 . Let M( ξ) be the infinite cyclic cover corresponding to ξ, with generating covering translation t: M( ξ)→ M( ξ). Let M(ξ, k) be the quotient M( ξ)/ t k . We prove that lim k→∞ M(M(ξ, k))/k exists. For ξ outside a subset M⊂H 1(M) which is the union of a finite family of hyperplanes, we obtain the asymptotics of M(M(ξ, k)) as k→∞ in terms of homotopy invariants of M related to the Novikov homology of M. It turns out that the limit above does not depend on ξ (if ξ∉ M ). Similar results hold for the stable Morse numbers. Generalizations for the case of non-cyclic coverings are obtained.

Knot invariants from symbolic dynamical systems

If G G is the group of an oriented knot k k , then the set Hom ⁡ ( K , Σ ) \operatorname {Hom} (K, \Sigma ) of representations of the commutator subgroup K = [ G , G ] K = [G,G] into any finite group Σ \Sigma has the structure of a shift of finite type Φ Σ \Phi _{\Sigma } , a special type of dynamical system completely described by a finite directed graph. Invariants of Φ Σ \Phi _{\Sigma } , such as its topological entropy or the number of its periodic points of a given period, determine invariants of the knot. When Σ \Sigma is abelian, Φ Σ \Phi _{\Sigma } gives information about the infinite cyclic cover and the various branched cyclic covers of k k . Similar techniques are applied to oriented links.

Surface singularities on cyclic coverings and an inequality for the signature

For the signature of the Milnor fiber of a surface singularity of cyclic type, we prove a certain inequality, which naturally induce an answer of Durfee's conjecture in this case. For the proof, we use a certain perturbation method on the way of Hirzebruch's resolution process.

On cyclic bra nched coverings of torus knots

We prove that then-fold cyclic coverings of the 3-sphere branched over the torus knotsK(p,q), p>q≥2 (i.e. the Brieskorn manifolds in the sense of [12]) admit spines corresponding to cyclic presentations of groups ifp≡1 (modq). These presentations include as a very particular case the Sieradski groups, first introduced in [14] and successively obtained from geometric constructions in [4], [9], and [15]. So our main theorem answers in affirmative to an open question suggested by the referee in [14]. Then we discuss a question concerning cyclic presentations of groups and Alexander polynomials of knots.

Homologically trivial actions on cyclic coverings of knots

Homologically trivial actions on cyclic coverings of knots

Hodge numbers attached to a polynomial map

We attach a limit mixed Hodge structure to any polynomial map f:ℂ n →ℂ. The equivariant Hodge numbers of this mixed Hodge structure are invariants of f which reflect its asymptotic behaviour. We compute them for a generic class of polynomials in terms of equivariant Hodge numbers attached to isolated hypersurface singularities and equivariant Hodge numbers of cyclic coverings of projective space branched along a hypersurface. We show how these invariants allow to determine topological invariants of f such as the real Seifert form at infinity.

Models of Curves and Finite Covers

Let K be a discrete valuation field with ring of integers OK .L etf :X! Y be afi nite morphism of curves overK. In this article, we study some possible relationships between the models over OK ofX and ofY . Three such relationships are listed below. Consider a Galois coverf :X!Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable reduction over K ,t henX achieves semi-stable reduction over some explicit tame extension of K.B/ .W henK is strictly henselian, we determine the minimal extensionL=K with the property thatXL has semi-stable reduction. Let f :X ! Y be a finite morphism, with g.Y/> 2. We show that if X has a stable model X over OK ,t henY has a stable model Y over OK , and the morphism f extends to a morphism X! Y. Finally, given any finite morphism f :X! Y , is it possible to choose suitable regular models X and Y of X and Y over OK such that f extends to a finite morphism X! Y ?A s was shown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situ- ations, withf a cyclic cover of any order> 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3. Mathematics Subject Classifications ( 1991): 14G20, 11G20, 14H30, 14H25. Let OK be a Dedekind domain with field of fractions K.L etf :X ! Y be a finite morphism of projective, smooth, and geometrically connected curves over Spec.K/. In this paper, we study some possible relationships between the models of X and of Y. In the first part of the paper, we look at semi-stable and stable models, while in the second part we investigate regular models. Let us describe the content of this paper. Definitions and standard facts about models are reviewed in the first section. Let B Y denote the branch locus of f ,a nd letK.B/ be the compositum, in an algebraic closure of K, of the residue fields of points of B. In the second section, we consider a Galois cover X! Y of degree prime top and show that, ifY has semi-stable reduction over K ,t henX achieves semi-stable reduction over an explicit tame extension of the field K.B/ (Theorem 2.3). When K is strictly Henselian, there exist extensions LX=K and

K3 Surfaces with Nine Cusps

By a K3-surface with nine cusps I mean a surface with nine isolated double points A2, but otherwise smooth, such that its minimal desingularisation is a K3-surface. It is shown that such a surface admits a cyclic triple cover branched precisely over the cusps. This parallels the theorem of Nikulin that a K3-surface with 16 nodes is a Kummer quotient of a complex torus.

Invariants for one-dimensional cohomology classes arising from TQFT

Let ( V, Z) be a Topological Quantum Field Theory over a field f defined on a cobordism category whose morphisms are oriented n + 1-manifolds perhaps with extra structure (for example a p 1 structure and banded link). Let ( M, χ) be a closed oriented n + 1-manifold M with this extra structure together with χ ϵ H 1( M). Suppose χ: H 1( M) → Z is an epimorphism. Let M ∞ denote the infinite cyclic cover of M given by χ. Consider a fundamental domain E for the action of the integers on M ∞ bounded by lifts of a surface Σ dual to χ, and in general position. E can be viewed as a cobordism from Σ to itself. We give Turaev and Viro's proof of their theorem that the similarity class of the nonnilpotent part of Z( E) is an invariant. We give a method to calculate this invariant for the ( V p , Z p ) theories of Blanchet, Habegger, Masbaum and Vogel when M is 0-framed surgery to S 3 along a knot K. We give a formula for this invariant when K is a twisted double of another knot. We obtain formulas for the quantum invariants of branched covers of knots, and unbranched covers of 0-surgery to S 3 along knots. We study periodicity among the quantum invariants of Brieskorn manifolds. We give an upper bound on the quantum invariants of branched covers of fibered knots. We also define finer invariants for pairs ( M, χ) for TQFTs over Dedekind domains. We use these ideas to study isotopy invariants of banded links in S 1 × S 2.

Autamorphisms of generic cyclic covers

We generalize an argument of LMa2] for proving a result abeut automorphisms of generic simple cyclic covers of smoetb aigebraic varieties. A finite map Ir: 5 —4 X is called a simple cychic cever if there exists an invertible sheaf C en X such that ,r.Os = e~jC’ (cf. [B-P-V] 1.17). Here we prove under sorne ‘rnild” assumptions en the triple X,4 vi that for the generic cyclic cover 5 the group Ant(S) of biregular automorphisms equals the group p,, of automorphisms of tbe brariched cover Ir.

Cyclic coverings of an elliptic curve with two branch points and the gap sequences at the ramification points

Cyclic coverings of an elliptic curve with two branch points and the gap sequences at the ramification points

Random Walks on Regular and Irregular Graphs

For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list until it completes a full cycle. The main result of this paper is a characterization of the cyclic cover time in terms of simple and easy-to-compute graph properties. Namely, for any connected graph, the cyclic cover time is $\Theta (n^2 d_{ave} (d^{ - 1} )_{ave} $), where n is the number of vertices in the graph, $d_{ave} $ is the average degree of its vertices, and $(d^{ - 1} )_{ave} $ is the average of the inverse of the degree of its vertices. Other results obtained in the processes of proving the main theorem are a similar characterization of minimum resistance spanning trees of graphs, improved bounds on the cover time of graphs, and a simplified proof that the maximum commute time in any connected graph is at most $4n^3 /27 + o(n^3 )$.

ALL LINS-MANDEL SPACES ARE BRANCHED CYCLIC COVERINGS OF S3

In this paper we show that all Lins-Mandel spaces S (b, l, t, c) are branched cyclic coverings of the 3-sphere. When the space is a 3-manifold, the branching set of the covering is a two-bridge knot or link of type (l, t) and otherwise is a graph with two vertices joined by three edges (a θ-graph). In the latter case the singular set of the space is always composed by two points with homeomorphic links. The first homology groups of the Lins-Mandel manifolds are computed when t=1 and when the branching set is a knot of genus one. Furthermore the family of spaces has been extended in order to contain all branched cyclic coverings of two-bridge knots or links.

On plane algebraic curves of positive Albanese dimension

A notion of Albanese dimension of a plane algebraic curve is introduced as the maximum of the dimensions of the images of Albanese mapping of cyclic coverings of the plane ramified along this curve. A necessary condition for a curve to have Albanese dimension equal to one is given in terms of its equation. This condition is sufficient for a generic curve. A complete description of the Alexander polynomials of plane curves of Albanese dimension one is given. Also the images of the Albanese mappings of cyclic coverings ramified along such curves are described completely.

On the monodromy and mixed Hodge structure on cohomology of the infinite cyclic covering of the complement to a plane algebraic curve

The semisimplicity is proved of the Alexander automorphism (the monodromy operator) on the cohomology of the infinite cyclic covering of the complement to a plane non-reduced algebraic curve, and, in particular, the semisimplicity of in the case of an irreducible curve. A natural mixed Hodge structure on is introduced and the irregularity of cyclic coverings of is calculated in terms of the number of roots of the Alexander polynomial of the branch curve.