This paper aims to systematically treat the methods by which topological questions about compact Lie groups can be reduced to algebraic questions about Lie algebras. The reduction process involves three steps: (1) replacing questions on homology groups with questions on differential forms, (2) focusing on invariant differential forms using invariant integration on the group manifold, and (3) transforming these into alternating multilinear forms on the Lie algebra of the group. The first chapter introduces differential forms on manifolds and equivariant forms, which are forms defined in terms of a linear representation of the group. It discusses the properties of these forms and their relationship to each other. The second chapter deals with the case where the group acts transitively on the manifold, leading to the computation of Betti numbers of homogeneous spaces using algebraic methods. The third and fourth chapters focus on the cohomology groups of Lie algebras, derived from both invariant and equivariant forms. Properties of these cohomology groups are derived from both transcendental properties of compact groups and purely algebraic methods. The paper also includes theorems on the structure of semi-simple Lie algebras and the cohomology of compact connected semi-simple Lie groups. The paper concludes with a discussion on the averaging process and the localization of forms, providing a detailed analysis of the relationship between differential forms on Lie groups and their corresponding forms on the Lie algebra. It also explores the invariance properties of forms under left and right translations, leading to the characterization of invariant forms and their role in the cohomology of compact Lie groups.This paper aims to systematically treat the methods by which topological questions about compact Lie groups can be reduced to algebraic questions about Lie algebras. The reduction process involves three steps: (1) replacing questions on homology groups with questions on differential forms, (2) focusing on invariant differential forms using invariant integration on the group manifold, and (3) transforming these into alternating multilinear forms on the Lie algebra of the group. The first chapter introduces differential forms on manifolds and equivariant forms, which are forms defined in terms of a linear representation of the group. It discusses the properties of these forms and their relationship to each other. The second chapter deals with the case where the group acts transitively on the manifold, leading to the computation of Betti numbers of homogeneous spaces using algebraic methods. The third and fourth chapters focus on the cohomology groups of Lie algebras, derived from both invariant and equivariant forms. Properties of these cohomology groups are derived from both transcendental properties of compact groups and purely algebraic methods. The paper also includes theorems on the structure of semi-simple Lie algebras and the cohomology of compact connected semi-simple Lie groups. The paper concludes with a discussion on the averaging process and the localization of forms, providing a detailed analysis of the relationship between differential forms on Lie groups and their corresponding forms on the Lie algebra. It also explores the invariance properties of forms under left and right translations, leading to the characterization of invariant forms and their role in the cohomology of compact Lie groups.