Finite Cover Research Articles

On positive scalar curvature and moduli of curves

In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus $g$ with $g\geq 2$ does not admit any Riemannian metric $ds^2$ of nonnegative scalar curvature such that $ds^2 \succ ds_{T}^2$ where $ds_{T}^2$ is the Teichm\"uller metric. Our second result is the proof that any cover $M$ of the moduli space $\mathbb{M}_{g}$ of a closed Riemann surface $S_{g}$ does not admit any complete Riemannian metric of uniformly positive scalar curvature in the quasi-isometry class of the Teichm\"uller metric, which implies a conjecture of Farb-Weinberger.

Wild solenoids

A weak solenoid is a foliated space defined as the inverse limit of finite coverings of a closed compact manifold M M . The monodromy of a weak solenoid defines an equicontinuous minimal action on a Cantor space X X by the fundamental group G G of M M . The discriminant group of this action is an obstruction to this action being homogeneous. The discriminant vanishes if the group G G is abelian, but there are examples of actions of nilpotent groups for which the discriminant is non-trivial. The action is said to be stable if the discriminant group remains unchanged for the induced action on sufficiently small clopen neighborhoods in X X . If the discriminant group never stabilizes as the diameter of the clopen set U U tends to zero, then we say that the action is unstable, and the weak solenoid which defines it is said to be wild. In this work, we show two main results in the course of our study of the properties of the discriminant group for Cantor actions. First, the tail equivalence class of the sequence of discriminant groups obtained for the restricted action on a neighborhood basis system of a point in X X defines an invariant of the return equivalence class of the action, called the asymptotic discriminant, which is consequently an invariant of the homeomorphism class of the weak solenoid. Second, we construct uncountable collections of wild solenoids with pairwise distinct asymptotic discriminant invariants for a fixed base manifold M M , and hence fixed finitely-presented group G G , which are thus pairwise non-homeomorphic. The study in this work is the continuation of the seminal works on homeomorphisms of weak solenoids by Rogers and Tollefson in 1971 and is dedicated to the memory of Jim Rogers.

Irreducible fractal structures for Moran type theorems

In this paper, we introduce a separation property for self-similar sets which is necessary to reach the equality between the similarity dimension and the Hausdorff dimension of these spaces. The similarity boundary of a self-similar set is investigated from the viewpoint of that property. In this way, the strong open set condition (in the self-similar set setting) posed by Keesling and Krishnamurthi has been weakened leading to a Moran type theorem. Moreover, both a result based on a conjecture posed by Deng and Lau as well as an improved version of a theorem due to Bandt and Rao have been contributed regarding the size of the overlaps among the pieces of a self-similar set. Several (equivalent) conditions leading to the equality between the similarity dimension and a new Hausdorff type dimension for attractors described in terms of finite coverings are also provided. Finally, we list some open questions.

On Some Properties of Vector Measures

We study the properties of a parameterized sequence of countably additive vector measures with densities defined on a compact space T with a nonnegative nonatomic Radon measure μ and taking values in a separable Banach space. Each vector measure of this sequence depends continuously on a parameter belonging to a metric space. We assume that a countable locally finite open cover and a partition of unity inscribed into this cover are given in the metric space of parameters. We prove that, for each value of the parameter, there exists a sequence of μ-measurable subsets of the space T which is a partition of T. In addition, this sequence depends uniformly continuously on the parameter and, for each value of the parameter and each element of the initial parameterized sequence of vector measures, the relative value of the measure of the corresponding subset in the partition of T can be uniformly approximated by the corresponding value of the corresponding partition of unity function.

Two categorical characterizations of local fields

In the present paper, we discuss two categorical characterizations of local fields. We first prove that a certain full subcategory of the category of finite flat coverings of the spectrum of the ring of integers of a local field equipped with coherent modules completely determines the isomorphism class of the local field. Next, we also prove that a certain full subcategory of the category of irreducible schemes which are finite over the spectrum of the ring of integers of a local field completely determines the isomorphism class of the local field.

Group action on local dendrites

Let G be a group acting by homeomorphisms on a local dendrite X with countable set of endpoints. In this paper, it is shown that any minimal set M of G is either a finite orbit, or a Cantor set or a circle. Furthermore, we prove that if G is a finitely generated group, then the flow (G,X) is a pointwise recurrent flow if and only if one of the following two statements holds:(1)X=S1, and (G,S1) is a minimal flow conjugate to an isometric flow, or to a finite cover of a proximal flow;(2)(G,X) is pointwise periodic.

The free group does not have the finite cover property

We prove that the first order theory of nonabelian free groups eliminates the ∃∞-quantifier (in eq). Equivalently, since the theory of nonabelian free groups is stable, it does not have the finite cover property.

Frobenius Structures Over Hilbert C*-Modules

We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative and central dagger Frobenius structures. Subobjects of the tensor unit correspond to clopen subsets of the Gelfand spectrum of the C*-algebra, and we discuss dagger kernels.

Lifts of simple curves in finite regular coverings of closed surfaces

Suppose S is a closed orientable surface and $${\tilde{S}}$$ is a finite sheeted regular cover of S. When studying mapping class groups, the following question arose: do the lifts of simple curves from S generate $$H_{1}({\tilde{S}},{\mathbb {Z}})$$ ? A family of examples is given for which the answer is “no”.

The number of fiberings of a surface bundle over a surface

For a closed manifold $M$, let Fib$(M)$ be the number of distinct fiberings of $M$ as a fiber bundle with fiber a closed surface. In this paper we give the first computation of Fib$(M)$ where $1<\text{Fib}(M)<\infty$ but $M$ is not a product. In particular, we prove Fib$(M)=2$ for the Atiyah-Kodaira manifold and any finite cover of a trivial surface bundle. We also give an example where Fib$(M)=4$.

Galois covers of type $(p,\cdots,p)$, vanishing cycles formula, and the existence of torsor structures

In this article we prove a local Riemman-Hurwitz formula which compares the dimensions of the spaces of vanishing cycles in a finite Galois cover of type $(p,p,\cdots,p)$ between formal germs of $p$-adic curves and which generalises the formula proven in [6] in the case of Galois covers of degree $p$. We also investigate the problem of the existence of a torsor structure for a finite Galois cover of type $(p,p,\cdots,p)$ between $p$-adic schemes.

An Explicit Reciprocity Law Associated to Some Finite Coverings of Algebraic Curves

We provide a new reciprocity law associated with finite coverings of algebraic curves. Moreover, we give explicit examples of this new reciprocity law that are not trivial consequences of the Weil reciprocity law over the base curve.

Towers of regular self-covers and linear endomorphisms of tori

Let $M$ be a closed manifold that admits a self-cover $p:M \to M$ of degree >1. We say p is strongly regular if all its iterates are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of $M$: We prove that $\pi_1(M)$ surjects onto a nontrivial free abelian group $A$, and the self-cover is induced by a linear endomorphism of $A$. Under further hypotheses we show that a finite cover of $M$ admits the structure of a principal torus bundle. We show that this applies when $M$ is K\"ahler and $p$ is a strongly regular, holomorphic self-cover, and prove that a finite cover splits as a product with a torus factor.

Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

We prove that every smooth diffeomorphism group valued cocycle over certain$\mathbb{Z}^{k}$Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan$\mathbb{Z}^{k}$($k\geq 3$) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over$\mathbb{Z}^{k}$actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

Topology of holomorphic Lefschetz pencils on the four-torus

We discuss topological properties of holomorphic Lefschetz pencils on the four-torus. Relying on the theory of moduli spaces of polarized abelian surfaces, we first prove that, under some mild assumptions, the (smooth) isomorphism class of a holomorphic Lefschetz pencil on the four-torus is uniquely determined by its genus and divisibility. We then explicitly give a system of vanishing cycles of the genus- 3 holomorphic Lefschetz pencil on the four-torus due to Smith, and obtain those of holomorphic pencils with higher genera by taking finite unbranched coverings. One can also obtain the monodromy factorization associated with Smith’s pencil in a combinatorial way. This construction allows us to generalize Smith’s pencil to higher genera, which is a good source of pencils on the (topological) four-torus. As another application of the combinatorial construction, for any torus bundle over the torus with a section we construct a genus- 3 Lefschetz pencil whose total space is homeomorphic to that of the given bundle.

Étale endomorphisms of 3-folds. I

This paper is the first part of our project towards classifications of smooth projective $3$-folds $X$ with $\kappa(X) = -\infty$ admitting a non-isomorphic etale endomorphism. We can prove that for any extremal ray $R$ of divisorial type, the contraction morphism $\pi_R\colon X\to X'$ associated to $R$ is the blowing-up of a smooth $3$-fold $X'$ along an elliptic curve. The difficulty is that there may exist infinitely many extremal rays on $X$. Thus we introduce the notion of an `ESP' which is an infinite sequence of non-isomorphic finite etale coverings of $3$-folds with constant Picard number. We can run the minimal model program (`MMP') with respect to an ESP and obtain the `FESP' $Y_{\bullet}$ of $(X, f)$ which is a distinguished ESP with \textit{extremal rays of fiber type} (cf. Definition 3.6). We first classify $Y_{\bullet}$ and then blow-up $Y_{\bullet}$ along elliptic curves to recover the original $X$. The finiteness of extremal rays of $\overline{\rm NE}(X)$ is verified in certain cases (cf. Theorem 1.4). We encounter a new phenomenon showing that our \'{e}taleness assumption is related with torsion line bundles on an elliptic curve (cf. Theorem 1.5).

Some anomalous examples of lifting spaces

An inverse limit of a sequence of covering spaces over a given space X is not, in general, a covering space over X but is still a lifting space, i.e. a Hurewicz fibration with unique path lifting property. Of particular interest are inverse limits of finite coverings (resp. finite regular coverings), which yield fibrations whose fibre is homeomorphic to the Cantor set (resp. profinite topological group). To illustrate the breadth of the theory, we present in this note some curious examples of lifting spaces that cannot be obtained as inverse limits of covering spaces.

Epimorphisms of 3-manifold groups

Let f: M -> N be a proper map between two aspherical compact orientable 3-manifolds with empty or toroidal boundary. We assume that N is not a closed graph manifold. Suppose that f induces an epimorphism on fundamental groups. We show that f is homotopic to a homeomorphism if one of the following holds: either for any finite-index subgroup Gamma of pi(1)(N) the ranks of Gamma and of f(*)(-1) (Gamma) agree, or for any finite cover (N) over tilde of N the Heegaard genus of (N) over tilde and the Heegaard genus of the pull-back cover (M) over tilde agree.

Virtual 1-domination of 3-manifolds

It is shown in this paper that given any closed oriented hyperbolic 3-manifold, every closed oriented 3-manifold is mapped onto by a finite cover of that manifold via a map of degree 1, or in other words, virtually 1-dominated by that manifold. This improves a known result of virtual 2-domination. The proof invokes a recently developed enhanced version of the connection principle in good pants constructions.

Growth of homology torsion in finite coverings and hyperbolic volume

We give an upper bound for the growth of homology torsions of finite coverings of irreducible 3-manifolds with tori boundary in terms of hyperbolic volume.