Basic Algebraic Topology: the Fundamental Group of a Circle
نویسنده
چکیده
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 fundamental group of a circle, one of the most foundational calculations in algebraic topology.
منابع مشابه
Algebraic Topology (hatcher)
Part 1. The Fundamental Group 3 2. Basic Constructions 3 2.1. Paths and Homotopy 3 2.2. The Fundamental Group of the Circle 4 2.3. Induced Homomorphisms 6 3. Van Kampen’s Theorem 7 3.1. Free Products of Groups 7 3.2. The Van Kampen Theorem 7 3.3. Application to Cell Complexes 9 4. Covering Spaces 11 4.1. Lifting Properties 11 4.2. The Classification of Covering Spaces 12 4.3. Deck Transformatio...
متن کاملOn subgroups of topologized fundamental groups and generalized coverings
In this paper, we are interested in studying subgroups of topologized fundamental groups and their influences on generalized covering maps. More precisely, we find some relationships between generalized covering subgroups and the other famous subgroups of the fundamental group equipped with the compact-open topology and the whisker topology. Moreover, we present some conditions unde...
متن کاملAlgebraic and topological aspects of quasi-prime ideals
In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is pro...
متن کاملThe Fundamental Algebraic Group of Topological Spaces
In the study of topology, we are often interested in understanding and classifying the internal structure of topological spaces. Algebraic topology is the application of abstract algebra to topology in order to further identify the structure of topological spaces by developing a correspondence between topological spaces and certain groups called homotopy groups. In this paper, we will examine t...
متن کاملInternal Topology on MI-groups
An MI-group is an algebraic structure based on a generalization of the concept of a monoid that satisfies the cancellation laws and is endowed with an invertible anti-automorphism representing inversion. In this paper, a topology is defined on an MI-group $G$ under which $G$ is a topological MI-group. Then we will identify open, discrete and compact MI-subgroups. The connected components of th...
متن کامل