In the study of plane curves, one of the problems is to classify the embedded topology of plane curves in the complex projective plane that have a given fixed combinatorial type, where the combinatorial type of a plane curve is data equivalent to the embedded topology in its tubular neighborhood. A pair of plane curves with the same combinatorial type but distinct embedded topology is called a Zariski pair. In this paper, we consider Zariski pairs consisting of conic-line arrangements that arise from Poncelet’s closure theorem. We study unramified double covers of the union of two conics that are induced by a $2m$-sided Poncelet transverse. As an application, we show the existence of families of Zariski pairs of degree $2m+6$ for $m\geq 2$ that consist of reducible curves having two conics and $2m+2$ lines as irreducible components.

Let $p \;:\; Y \to X$ be a finite, regular cover of finite graphs with associated deck group $G$, and consider the first homology $H_1(Y;\;{\mathbb{C}})$ of the cover as a $G$-representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group $G$ on the one hand and topological properties of homology classes in $H_1(Y;\;{\mathbb{C}})$ on the other hand. We do so by studying certain subrepresentations in the $G$-representation $H_1(Y;\;{\mathbb{C}})$.

The homology class of a lift of a primitive element in $\pi _1(X)$ spans an induced subrepresentation in $H_1(Y;\;{\mathbb{C}})$, and we show that this property is never sufficient to characterize such homology classes if $G$ is Abelian. We study $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \leq H_1(Y;\;{\mathbb{C}})$—the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in $\pi _1(X)$. Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \neq \ker\!(p_*)$.

Every Anosov flow on a 3-manifold is associated to a bifoliated plane (a plane endowed with two transverse foliations $F^s$ and $F^u$ ) which reflects the normal structure of the flow endowed with the center-stable and center-unstable foliations. A flow is $\mathbb{R}$ -covered if $F^s$ (or equivalently $F^u$ ) is trivial. On the other hand, from any Anosov flow one can build infinitely many others by Dehn–Goodman–Fried surgeries. This paper investigates how these surgeries modify the bifoliated plane. We first observe that surgeries along orbits corresponding to disjoint simple closed geodesics do not affect the bifoliated plane of the geodesic flow of a hyperbolic surface (Theorem 1). Analogously, for any non- $\mathbb{R}$ -covered Anosov flow, surgeries along pivot periodic orbits do not affect the branching structure of its bifoliated plane (Theorem 2). Next, we consider the set $\mathcal{S}urg(A)$ of Anosov flows obtained by Dehn–Goodman–Fried surgeries from the suspension flow $X_A$ of any hyperbolic matrix $A \in SL(2,\mathbb{Z})$ . Fenley proved that performing only positive (or negative) surgeries on $X_A$ leads to $\mathbb{R}$ -covered Anosov flows. We study here Anosov flows obtained by a combination of positive and negative surgeries on $X_A$ . Among other results, we build non- $\mathbb{R}$ -covered Anosov flows on hyperbolic manifolds. Furthermore, we show that given any flow $X\in \mathcal{S}urg(A)$ there exists $\epsilon>0$ such that every flow obtained from $X$ by a non-trivial surgery along any $\epsilon$ -dense periodic orbit $\gamma$ is $\mathbb{R}$ -covered (Theorem 4). Analogously, for any flow $X \in \mathcal{S}urg(A)$ there exist periodic orbits $\gamma_+,\gamma_-$ such that every flow obtained from $X$ by surgeries with distinct signs on $\gamma_+$ and $\gamma_-$ is non- $\mathbb{R}$ -covered (Theorem 5).

In this paper we apply Conley index theory in a covering space of an invariant set S, possibly not isolated, in order to describe the dynamics in S. More specifically, we consider the action of the covering translation group in order to define a topological separation of S which distinguishes the connections between the Morse sets within a Morse decomposition of S. The theory developed herein generalizes the classical connection matrix theory, since one obtains enriched information on the connection maps for non-isolated invariant sets, as well as for isolated invariant sets. Moreover, in the case of an infinite cyclic covering induced by a circle-valued Morse function, one proves that the Novikov differential of f is a particular case of the p-connection matrix defined herein.

If $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ is a covering map between connected graphs, and $H$ is the subgroup of $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is isomorphic to $N(H)/H$, where $N(H)$ is the normalizer of $H$ in $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$. We show that an entirely analogous result holds for immersions between connected graphs, where the subgroup $H$ is replaced by the closed inverse submonoid of the inverse monoid $L(\unicode[STIX]{x1D6E4},v)$ used to construct the immersion. We observe a relationship between group actions on graphs and deck transformations of graph immersions. We also show that a graph immersion $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ may be extended to a cover $g:\tilde{\unicode[STIX]{x1D6E5}}\rightarrow \unicode[STIX]{x1D6E4}$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$.

Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

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.

In a study of the word problem for groups, R. J. Thompson considered a certain group $F$ of self-homeomorphisms of the Cantor set and showed, among other things, that $F$ is finitely presented. Using results of K. S. Brown and R. Geoghegan, M. N. Dyer showed that $F$ is the fundamental group of a finite two-complex ${{Z}^{2}}$ having Euler characteristic one and which is Cockcroft, in the sense that each map of the two-sphere into ${{Z}^{2}}$ is homologically trivial. We show that no proper covering complex of ${{Z}^{2}}$ is Cockcroft. A general result on Cockcroft properties implies that no proper regular covering complex of any finite two-complex with fundamental group $F$ is Cockcroft.

If $V\,\to \,X$ is a vector bundle of fiber dimension $k$ and $Y\,\to \,X$ is a finite sheeted covering map of degree $d$, the implications for the Euler class $e(V)$ in ${{H}^{k}}(X)$ of $V$ implied by the existence of an embedding $Y\,\to \,V$ lifting the covering map are explored. In particular it is proved that $d{{d}^{\prime }}\text{e(V)}\text{=}\text{0}$ where ${{d}^{\prime }}$ is a certain divisor of $d\,-\,1$, and often ${{d}^{\prime }}=1$.