Simply-connected K-contact and Sasakian manifolds of dimension 7

TL;DR

This paper constructs a specific 7-dimensional manifold with a K-contact structure that is not Sasakian, and explores its rational homotopy properties, revealing conditions for formality and cohomology cup-products.

Contribution

It provides the first example of a simply-connected 7-dimensional K-contact manifold that is not Sasakian and analyzes its rational homotopy characteristics.

Findings

Existence of a simply-connected 7-dimensional K-contact manifold that is not Sasakian.

A simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on H^2.

Such Sasakian manifolds are formal iff all triple Massey products vanish.

Abstract

We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and that it is formal if and only if all its triple Massey products vanish.