The aim of this work is to build an intersection product in homology classes of a smooth compact orientable manifold so that this intersection is Poincaré dual of cup product of its cohomology classes. To do this, we start from the definition of intersection in complementary grades, and we use the intersection number from differential topology plus stratified transversality theorem to extend this definition to multiple intersection. Then, through a convenient version of Poincaré Duality, we obtain a definition of intersection in any grades. Finally, we calculate the intersection algebra of separating Dehn twist and Johnsons twist.