On the formality of nearly K\"ahler manifolds and of Joyce's examples in $G_2$-holonomy

TL;DR

This paper investigates the formality of manifolds with special holonomy, proving nearly Kähler manifolds are formal and providing a method to confirm the formality of Joyce's $G_2$-holonomy examples, advancing understanding in rational homotopy theory.

Contribution

It establishes the formality of nearly Kähler manifolds and introduces a concrete method to verify the formality of Joyce's $G_2$ examples, aiding future research in special holonomy spaces.

Findings

Nearly Kähler manifolds are formal.

A method to confirm the formality of Joyce's $G_2$ examples is provided.

Potential extension to $Spin(7)$ holonomy cases.

Abstract

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 $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ 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 $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly K\"ahler manifolds. A connection between these two results can be found in the fact that both "special holonomy" and "nearly K\"ahler" naturally generalize compact K\"ahler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan.