The rational homotopy type of (n-1)-connected (4n-1)-manifolds

Diarmuid Crowley, Johannes Nordström

Published 2015-05-15Version 1

We define the Binachi-Massey tensor on the degree n cohomology with rational coefficients of a topological space X as a linear map from a subspace of the fourth tensor power of H^n(X) (determined by the cup product H^n(X) x H^n(X) -> H^{2n}(X)) to H^{4n-1}(X). If M is a closed (n-1)-connected (4n-1)-manifold (and n > 1) then its rational homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and 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.