This paper presents a result in 4-manifold topology, announced in [7], proved using analytical and geometrical methods. It shows that the homotopy type of a closed simply connected 4-manifold is determined by the cup-square, a quadratic form on the second cohomology group. Freedman showed that any such form can be realized by a simply connected topological 4-manifold, and that the form, together with the Kirby-Siebenmann obstruction, determines the manifold up to homeomorphism. However, some forms cannot be realized by smooth 4-manifolds. Rohlin's theorem states that even forms from smooth 4-manifolds have signature divisible by 16. The main theorem here shows that if the associated form Q is positive definite, it is equivalent to the standard diagonal form. This implies that certain manifolds constructed by Freedman cannot be given a differentiable structure. The paper also discusses the methods used to prove this result, including the use of gauge theory and self-dual connections. The proof involves constructing a moduli space of self-dual connections and showing that it is a smooth 5-manifold. The paper concludes with a theorem showing that there exists an open subset of the moduli space that is a smooth 5-manifold diffeomorphic to X × (0, λ₀), with the complement being compact. This completes the proof of the main theorem.This paper presents a result in 4-manifold topology, announced in [7], proved using analytical and geometrical methods. It shows that the homotopy type of a closed simply connected 4-manifold is determined by the cup-square, a quadratic form on the second cohomology group. Freedman showed that any such form can be realized by a simply connected topological 4-manifold, and that the form, together with the Kirby-Siebenmann obstruction, determines the manifold up to homeomorphism. However, some forms cannot be realized by smooth 4-manifolds. Rohlin's theorem states that even forms from smooth 4-manifolds have signature divisible by 16. The main theorem here shows that if the associated form Q is positive definite, it is equivalent to the standard diagonal form. This implies that certain manifolds constructed by Freedman cannot be given a differentiable structure. The paper also discusses the methods used to prove this result, including the use of gauge theory and self-dual connections. The proof involves constructing a moduli space of self-dual connections and showing that it is a smooth 5-manifold. The paper concludes with a theorem showing that there exists an open subset of the moduli space that is a smooth 5-manifold diffeomorphic to X × (0, λ₀), with the complement being compact. This completes the proof of the main theorem.