M.Phil / MS
Permanent URI for this collectionhttps://repository.cuilahore.edu.pk/handle/123456789/52
This collection archives the complete set of theses produced by students of the COMSATS University Islamabad, Lahore Campus.
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Item On Spanning Simplicial Complexes Associated to Ladder Graph(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) MUHAMMAD AZIZ; CIIT/FA19-RMT-011/LHR; Dr. Hani Shaker; LHR TP 7427In this thesis, we are going to discuss the algebraic and combinatorial aspects of span- ning simplicial complex ∆s(G) associated with the simple connected graph G, namely Ladder graph Ln. We mainly emphasis the characterization of s(Ln), all spanning tree, of Ln. We intend to provide the algebraic and combinatorial characterization of SSC ∆s(Ln) associated with the ladder graph Ln. In particular, we compute the formula for the f -vector of the SSC associated with the ladder graph Ln.Item Fundamental Group and Singular Homology(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) DANIAL HUSSAIN; CIIT/FA19-RMT-008/LHR; Dr. Imran Ahmed; LHR TP 7442Algebraic topology is a very fascinating and beautiful subject. In this area of math- ematics, we study shapes, where we are interested in what is maintained when we continuously deform shapes. The main objective of my thesis is to introduce the basic notions of algebraic topology, such as fundamental group π1 and singular homology group H1(these groups are isomorphic for homotopically equivalent topo- logical spaces). My thesis work is self-contained and explanatory. In 2nd chapter, I have discussed the “fundamental group, its properties, and some very important theorems like the Borsuk-Ulam theorem, Brouwer fixed-point theorem, and Van- Kampen’s theorem”. While “singular homology, its basic properties, homology of a point, induced map between chain complexes, pushforward of homology, homotopy invariance, relative homology, long exact sequence property, induced maps, exci- sion theorem, homology of a quotient, and relation between H1 and π1” have been discussed in chapter 3.Item The Fundamental Groups and Classification of Covering Spaces(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Shoaib Khan; CIIT/FA19-RMT-050/LHR; Dr. Hani Shaker; LHR TP 7439Algebraic topology is a very rich subject and its beauty lies in the way chosen to study space. Based on the global properties of spaces, the constructions and developments in this subject give an abstract and general notion. Defining algebraic topology as the study of topological spaces by using the algebraic invariants is the very basic interpretation of algebraic topology. However, the fundamental group is one of these topological invariants used to study topological spaces. The algebraic variants such as fundamental group are used for the classification of topological spaces up to the homeomorphism. However, most of the invariants classify topological spaces up to homotopy equivalence. The general idea of algebraic topology and fundamental group has been given in the 1st chapter. In this thesis work, to explain fundamental group, homeomorphism, and homotopy, the 2nd chapter includes the gathered ideas and concepts of homotopy of loops and maps, fundamental group, and induced homeomorphism. This chapter forms a basis for covering spaces which has a direct link with the fundamental group. After setting the basis for covering spaces, the 3rd chapter explains all the aspects related to covering spaces such as the lifting of maps to the covering space, and the applications of the fundamental group are explained through the major algebraic topology theorems; “Brouwer Fixed Point Theorem” and “Borsuk Ulam Theorem”. A main result named as “Van-Kampan Theorem” has also been proved which helps a lot to calculate the fundamental groups of different spaces. In chapter 4 all the possible covering spaces of the given topological space have been discussed leading to the classification of covering spaces. Moreover, this section discusses the classification of covering spaces by starting from “The Covering Transformation”, “Galois Covering Space”, “Universal Cover” and “Galois Correspondence”.