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

TL;DR

This paper introduces the Bianchi-Massey tensor to classify the rational homotopy types of certain high-dimensional, highly connected manifolds, establishing formality criteria and providing new classification results.

Contribution

It defines the Bianchi-Massey tensor and proves it, along with cohomology, determines the rational homotopy type of (n-1)-connected manifolds up to dimension 5n-3, including formality conditions.

Findings

The Bianchi-Massey tensor determines the rational homotopy type of the manifolds.

Formality of the manifold is equivalent to the vanishing of the Bianchi-Massey tensor.

Classification of simply-connected 7-manifolds up to finite ambiguity.

Abstract

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 rational 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.