Let X̃ be a connected space, X be a space, let p : X̃ −→ X be a continuous map and let (X̃, p) be a covering space of X . In the first section we give some preliminaries from covering spaces and their automorphism groups. In the second section we derive some algebraic properties of both universal and regular covering spaces (X̃, p) of X and also their automorphism groups A(X̃, p). Keywords—covering ...

In this paper we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G1 and G2, if V (H) = V (G1) × V (G2) and both projection maps V (H) → V (G1) and V (H) → V (G2) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP -hard to decide). We investigate the conditions unde...

In this paper we present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. We show that the graphs in all such families possess some general properties including regularity and bi-regularity, existence of special vertex colorings, and existence of covering maps — hence, embedded spectra — between every two members of ...

In this paper we prove sharp covering theorems for nonconstant holomorphic functions f in the unit disk U. Theorem 1 asserts that if |f ′(0)| ≥ A|f(0)|, where A is a given number larger than 4, then f covers some annulus of the form r < |w| < Kr, where K = K(A) is a number depending on A. The theorem is sharp; extremals are furnished by universal covering maps from U onto the plane minus a doub...

Making use of a modified version, due to McCord, of the Dold-Thom construction of ordinary homology, we give a simple topological definition of a transfer for ramified covering maps in homology with arbitrary coefficients. The transfer is induced by a suitable map between topological groups. We also define a new cohomology transfer which is dual to the homology transfer. This duality allows us ...

The main purpose of this paper is to popularize Danzer’s power complex construction and establish some new results about covering maps between two power complexes. Power complexes are cube-like combinatorial structures that share many structural properties with higher-dimensional cubes and cubical tessellations on manifolds. Power complexes that are also abstract polytopes have repeatedly appea...

We construct an infinite family of (q + 1)−regular Ramanujan graphs Xn of girth 1. We also give covering maps Xn+1 → Xn such that the minimal common covering of all the Xn’s is the universal covering tree.

A self-map, either single-valued or multi-valued, is multiply fixed if every map homotopic to it has at least two fixed points. Let p : X̃ → X be a finite covering space, of degree n, of a connected finite polyhedron, and let f : X → X be a map. We lift f to an n-valued map φp,f : X̃ ( X̃ and prove that it is multiply fixed if the Nielsen number of f is greater than or equal to two. We obtain a fo...

A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all maps of groupoid structure are continuous. The notion of monodromy groupoid of a topological groupoid generalizes those of fundamental groupoid and universal cover. It was earlier proved that the monodromy groupoid o...

In this paper we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G1 and G2, if V (H) = V (G1) × V (G2) and both projection maps V (H) → V (G1) and V (H) → V (G2) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP -hard to decide). We investigate the conditions unde...