Rational homotopy and simply-connected 8-manifolds

TL;DR

This paper introduces a new rational homotopy invariant for topological spaces, linking formality of certain manifolds to the vanishing of this invariant and classifying specific 8-manifolds up to torsion.

Contribution

It defines a new invariant P and establishes its role in characterizing formality and classifying simply-connected 8-manifolds with specific cohomology.

Findings

Formality of certain manifolds is equivalent to the vanishing of P and the Bianchi-Massey tensor.

Elements of the group of specific 8-manifolds are determined by P up to torsion.

The invariant P provides a new tool for understanding the topology of 8-manifolds.

Abstract

We introduce a rational homotopy invariant P of a topological space, which is a quintic tensor on the cohomology. For n > 1, formality of a closed (n-1)-connected manifold of dimension up to 5n-2 is equivalent to vanishing of P and the Bianchi-Massey tensor introduced by Crowley and the second author arXiv:1505.04184. We show also that elements of the group of closed simply-connected spin 8-manifolds with the cohomology of an r-fold connected sum of S^2 x S^6 are determined up to torsion by the value of P.