We define the Bianchi-Massey tensor of a topological space X to be a linear map from a subquotient of the fourth tensor power of H*(X). We then prove that if M is a closed (n-1)-connected manifold of dimension at most 5n-3 (and n > 1) then its ration
al homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and that M is formal if and only if the Bianchi-Massey tensor vanishes. We use the Bianchi-Massey tensor to show that there are many (n-1)-connected (4n-1)-manifolds that are not formal but have no non-zero Massey products, and to present a classification of simply-connected 7-manifolds up to finite ambiguity.
We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first authors thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any
closed spin 7-manifold, regardless of whether the spin characteristic class p_M in the fourth integral cohomology of M is torsion. In addition we determine the inertia group of 2-connected M - equivalently the number of oriented smooth structures on the underlying topological manifold - in terms of p_M and the torsion linking form.
