On the formality of nearly Kähler manifolds and of Joyce's examples in $\mathrm{G}_2$-holonomy

It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already — an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $\mathrm{G}_2$ and $\mathrm{Spin}(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $\mathrm{G}_2$ indeed are formal spaces; we concretely exert this computation for one example, which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $\mathrm{Spin}(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kähler manifolds. A connection between these two results can be found in the fact that both “special holonomy” and “nearly Kähler” naturally generalize compact Kähler manifolds, whose formality is a classical and celebrated theorem by Deligne–Griffiths–Morgan–Sullivan.