The goal of this paper is to explore basic topics in Algebraic Topology. The paper will first begin by exploring the concept of loops on topological spaces and how one can look at these loops and form a group. From there, it moves to discuss covering spaces giving basic definitions and constructing the universal cover. After establishing this, covering spaces are used to calculate the fundament...

The model of the torus as a parallelogram in the plane with opposite sides identiied enables us to deene two families of parallel lines and to tessellate the torus, then to associate to each tessellation a toroidal map with an upward drawing. It is proved that a toroidal map has a tessellation representation if and only if its universal cover is 2-connected. Those graphs that admit such an embe...

A complete list of homogeneous operators in the Cowen Douglas class Bn(D) is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc.

Chain complexes with a group of operators are a well known tool in algebraic topology, where they arise naturally as the chain complex C∗X̃ of cellular chains of the universal cover X̃ of a reduced CW -complex X. The group of operators here is the fundamental group of X. J.H.C. Whitehead in his classical but little read paper [31] showed that the chain complex C∗(X̃) is useful for the homotopy cla...

Let X(∆) be the real toric variety associated to a smooth fan ∆. The main purpose of this article is: (i) to determine the fundamental group and the universal cover of X(∆), (ii) to give necessary and sufficient conditions on ∆ under which π1(X(∆)) is abelian, (iii) to give necessary and sufficient conditions on ∆ under which X(∆) is aspherical, and when ∆ is complete, (iv) to give necessary an...

Let Γ be an extended tilde geometry of rank n > 2 such that there exists a 1-covering γ : Γ → Φ where Φ is a c.Cn−1-geometry with orders 1, 2, . . . , 2. Suppose that the normalizer in Aut(Γ ) of the deck group of γ acts flag-transitively on Γ . We prove that, under these hypotheses, only three possibilities exist for the universal cover of Γ . c © 2007 Elsevier Ltd. All rights reserved.

Suppose P is a partially ordered set that is locally finite, has a least element, and admits a rank function. We call P a weighted-relation poset if all the covering relations of P are assigned a positive integer weight. We develop a theory of covering maps for weighted-relation posets, and in particular show that any weighted-relation poset P has a universal cover P̃ → P , unique up to isomorph...

We present a simple constructive proof of the Uniformization Theorem which works for plane domains. The proof is a combination of covering space theory and Koebe's constructive proof of the Riemann mapping theorem, and the resulting algorithm can be used to estimate the Poincar6 metric for the domain.

Many mathematicians encounter k-to-1 maps only in the study of covering maps. But, of course, k-to-1 maps do not have to be open. This paper touches on covering maps, and simple maps, but concentrates on ordinary k-to-1 functions (both continuous and finitely discontinuous) from one metric continuum to another. New results, old results, and ideas for further research are given; and a baker’s do...