Finite Sheeted Cover Research Articles

Cusp and b_1 growth for ball quotients and maps onto Z with finitely generated kernel

Let $M = \mathbb{B}^2 / \Gamma$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ have little to do with one another, in contrast with the well-understood cases of hyperbolic $2$- and $3$-manifolds. We also discuss growth of $b_1$ for congruence arithmetic lattices acting on $\mathbb{B}^2$ and $\mathbb{B}^3$. Along the way, we provide an explicit example of a lattice in $\mathrm{PU}(2, 1)$ admitting a homomorphism onto $\mathbb{Z}$ with finitely generated kernel. Moreover, we show that any cocompact arithmetic lattice $\Gamma \subset \mathrm{PU}(n, 1)$ of simplest type contains a finite index subgroup with this property.

Classification of smooth structures on a homotopy complex projective space

We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective n-space \(\mathbb {C}\textbf {P}^{n}\), where n=3 and 4. Let M2n be a closed smooth 2n-manifold homotopy equivalent to \(\mathbb {C}\textbf {P}^{n}\). We show that, up to diffeomorphism, M6 has a unique differentiable structure and M8 has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover N2n of \(\mathbb {C}\textbf {P}^{n}\) for n=4,7 or 8 and six distinct differentiable structures on N10.

Characters and coverings of compact groups

We consider characters and finite-sheeted coverings of compact connected Abelian groups and prove analytic and algebraic properties of characters. As an application of these results, we show that the character group of a compact connected Abelian group with trivial finite-sheeted coverings is divisible.

On multiplicity of maps

We consider classical (homotopy) root problem and its “global” version. In the paper, these problems are solved for finite-sheeted coverings of manifold-like continua. In particular case, namely of compact connected finite-dimensional abelian groups, we obtain solution for arbitrary continuous maps.

Fundamental groups and finite sheeted coverings

It is well known that for a connected locally path-connected semi-locally 1-connected space X , there exists a bi-unique correspondence between the pointed d -fold connected coverings and the transitive representations of the fundamental group of X in the symmetric group Σ d of degree d . The classification problem becomes more difficult if X is a more general space, particularly if X is not locally connected. In attempt to solve the problem for general spaces, several notions of coverings have been introduced, for example, those given by Lubkin or by Fox. On the other hand, different notions of ‘fundamental group’ have appeared in the mathematical literature, for instance, the Brown–Grossman–Quigley fundamental group, the Čech–Borsuk fundamental group, the Steenrod–Quigley fundamental group, the fundamental profinite group or the fundamental localic group. The main result of this paper determines different ‘fundamental groups’ that can be used to classify pointed finite sheeted connected coverings of a given space X depending on topological properties of X .

Criteria for virtual fibering

We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber dodecahedral space and the Bianchi orbifolds. Moreover, we show that a taut sutured compression body has a finite-sheeted cover with a depth one taut-oriented foliation.

ALGORITHMIC CONSTRUCTION OF KIRBY DIAGRAMS FOR BRANCHED COVERS

We present an algorithm that converts any description of a 3-manifold as a finite-sheeted cover of the 3-sphere, branched over some knot or knotted graph, into a Kirby diagram for that manifold, thus resolving the problem posed in [4].

Spaces whose finite sheeted covers are homeomorphic to a fixed space

Finitely generated groups that have the property that all their finite index subgroups are free Abelian are shown to be either free Abelian themselves or to have prime cyclic first homology. This group theoretic result allows one to show that the first homology of a finite connected cell complex that has the property that all of its non-trivial finite index covers have total space homeomorphic to a given space must be either cyclic of prime order or free Abelian. Other topological corollaries include a classification of such 2-complexes and a classification of compact 3-manifolds that have a non-trivial finite sheeted cover and have the property that all their finite-sheeted covers are themselves.

A classification of [formula omitted]-connected 2-complexes

The finite regular 2-complexes that have the property that every finite sheeted cover has total space homeomorphic to the base space are classified.

Tight contact structures on solid tori

In this paper we study properties of tight contact structures on solid tori. In particular we discuss ways of distinguishing two solid tori with tight contact structures. We also give examples of unusual tight contact structures on solid tori. We prove the existence of a Z \mathbb {Z} -valued and a R / 2 π Z \mathbb {R}/2\pi \mathbb {Z} -valued invariant of a closed solid torus. We call them the self-linking number and the rotation number respectively. We then extend these definitions to the case of an open solid torus. We show that these invariants exhibit certain monotonicity properties with respect to inclusion. We also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic. At the same time we discuss various ways of constructing a tight contact structure on a solid torus. We then produce examples of solid tori with tight contact structures and calculate self-linking and rotation numbers for these tori. These examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori. At the end, we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite-sheeted cover of that solid torus is no longer tight. This answers negatively a question asked by Eliashberg in 1990. We also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure, another example of unexpected behavior.

Computing self-intersections of closed geodesics on finite-sheeted covers of the modular surface

An algorithm is given for deciding whether a closed geodesic on a finite-sheeted cover of the modular surface has self-intersections; if it does, the algorithm gives them in the order they occur in traversing the geodesic. The following general result on geodesics is proved: any closed geodesic on a Riemann surface R can be lifted to a simple closed geodesic on some finite-sheeted cover of R. In the last two sections the connection with the stabilizer (under the modular group) of a Markov quadratic irrationality is discussed.

Parts as Finite-Sheeted Coverings of the Disk

Parts as Finite-Sheeted Coverings of the Disk