This paper presents a detailed proof of a result in 4-manifold topology, specifically addressing the homotopy type of closed, simply connected 4-manifolds. The key theorem states that if a 4-manifold \(X\) has a positive definite intersection form, then this form is equivalent to the standard diagonal form over the integers. This implies that the manifold is diffeomorphic to a product of spheres and complex projective spaces. The proof involves advanced techniques from gauge theory and differential geometry. It begins by constructing a space \(\mathfrak{M}^o(X)\) associated with \(X\), which can be compactified to an orientable 5-manifold with boundary \(X\) and singularities. The main tool is the concept of a self-dual connection on a principal \(SU(2)\) bundle over \(X\). Self-dual connections are solutions to the Yang-Mills equations with self-dual curvature, and they play a crucial role in the classification of 4-manifolds. The author uses the moduli space of self-dual connections, denoted \(\mathfrak{M}(X)\), to analyze the structure of \(X\). By perturbing the self-duality equations, the author shows that the moduli space can be compactified to a smooth 5-manifold with boundary \(X\). This compactification is achieved by perturbing the equations to create a smooth 5-manifold with finitely many singular points, ensuring that the moduli space is orientable and diffeomorphic to \(X \times (0, \lambda_0)\) for some \(\lambda_0 > 0\). The proof is divided into several parts, including the convergence of sequences of connections over the 4-ball and over \(X\), the definition of the compactified moduli space, and the verification that the projection map from the compactified moduli space to \(X\) is a diffeomorphism. The results are significant for understanding the classification of 4-manifolds and the role of gauge theory in topology.This paper presents a detailed proof of a result in 4-manifold topology, specifically addressing the homotopy type of closed, simply connected 4-manifolds. The key theorem states that if a 4-manifold \(X\) has a positive definite intersection form, then this form is equivalent to the standard diagonal form over the integers. This implies that the manifold is diffeomorphic to a product of spheres and complex projective spaces. The proof involves advanced techniques from gauge theory and differential geometry. It begins by constructing a space \(\mathfrak{M}^o(X)\) associated with \(X\), which can be compactified to an orientable 5-manifold with boundary \(X\) and singularities. The main tool is the concept of a self-dual connection on a principal \(SU(2)\) bundle over \(X\). Self-dual connections are solutions to the Yang-Mills equations with self-dual curvature, and they play a crucial role in the classification of 4-manifolds. The author uses the moduli space of self-dual connections, denoted \(\mathfrak{M}(X)\), to analyze the structure of \(X\). By perturbing the self-duality equations, the author shows that the moduli space can be compactified to a smooth 5-manifold with boundary \(X\). This compactification is achieved by perturbing the equations to create a smooth 5-manifold with finitely many singular points, ensuring that the moduli space is orientable and diffeomorphic to \(X \times (0, \lambda_0)\) for some \(\lambda_0 > 0\). The proof is divided into several parts, including the convergence of sequences of connections over the 4-ball and over \(X\), the definition of the compactified moduli space, and the verification that the projection map from the compactified moduli space to \(X\) is a diffeomorphism. The results are significant for understanding the classification of 4-manifolds and the role of gauge theory in topology.