Cohomology in algebra, geometry, physicsand statistics

Massey tensors and the rational homotopy type of 7- and 8-manifolds

Defining tensors on the cohomology of a differential graded algebra by multiplying Massey products by a further element gives an object that has less dependence on choices than the Massey products, making them easier to work with. On the other hand, all Massey products can be recovered from these tensors when the cohomology algebra satisfies Poincare duality, like for the de Rham complex of a closed manifold. Moreover, suitable interpretations of these tensors can capture information about the rational homotopy type even when the Massey products are undefined. For closed simply-connected 7-manifolds these tensors (along with the cohomology algebra itself) suffice to determine the rational homotopy type. Conjecturally the same is true for closed 8-manifolds, but at least the vanishing of the tensors is equivalent to formality in that case. This is based on joint work with Diarmuid Crowley and Csaba Nagy.
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