Fundamental Group and Singular Homology

Abstract

Algebraic 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.