On Poincaré conjecture and Gregory Perlman famous theorem.

Abstract

In this graduation thesis were titled *"On Poincaré Conjecture and Gregory Perelman’s Famous Theorem,"* we addressed fundamental concepts in algebraic topology and category theory. The first section covered omotopy theory**, examining the homotopy of continuous maps, contractible spaces, and a detailed analysis of the fundamental group and Poincaré group. The second section focused on **category theory**, exploring dual categories, subcategories, and specialized categories such as the category of pointed topological spaces and the category of chain complexes, while emphasizing the role of homotopical functors in connecting these mathematical structures. Together, these concepts provided the theoretical framework essential for derstanding the methods used to solve complex topological problems .