The rational homotopy type of (n−1)-connected manifolds of dimension up to 5n−3

Abstract

We define the Bianchi–Massey tensor of a topological space (Formula presented.) to be a linear map (Formula presented.), where (Formula presented.) is a subquotient of (Formula presented.) determined by the algebra (Formula presented.). We then prove that if (Formula presented.) is a closed (Formula presented.) -connected manifold of dimension at most (Formula presented.) (and (Formula presented.)) then its rational homotopy type is determined by its cohomology algebra and Bianchi–Massey tensor, and that (Formula presented.) is formal if and only if the Bianchi–Massey tensor vanishes. We use the Bianchi–Massey tensor to show that there are many (Formula presented.) -connected (Formula presented.) -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.