Classification of simply-connected 4-5 and 6-manifolds.

• The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the second homology group. Moreover a famous theorem of M. Freedman implies that the homeomorphism type of the manifold only depends on this intersection form, and on a $\mathbb{Z}/2\mathbb{Z}$-invariant, the Kirby–Siebenmann invariant
• Barden in 1965 has given the following classification result for simply-connected compact smooth manifolds. Let $M$ and $N$ be simply-connected, closed, smooth 5-manifolds and let $\phi:H_2(M)\cong H_2(N)$ be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then $\phi$ is realised by a diffeomorphism.
• A system of invariants is also known for simply-connected 6-manifolds, it is described in the paper "CUBIC FORMS AND COMPLEX 3-FOLDS" by Okonek and Van de Ven (L'enseignement mathématique, 1995).