Moscow Mathematical Journal

Volume 24, Issue 4, October–December 2024 pp. 495–512.

On the Formality of Nearly Kähler Manifolds and of Joyce's Examples in $\mathrm{G}_2$ -Holonomy

Authors: Manuel Amann (1) and Iskander A. Taimanov (2)
Author institution:(1) Institut für Mathematik, Differentialgeometrie, Universität Augsburg, Universitätsstraße 14, 86159 Augsburg, Germany
(2) Sobolev Institute of Mathematics, avenue academician Koptyug 4, 630090 Novosibirsk, Russia

Summary:

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.

2020 Math. Subj. Class. Primary: 53C29, 55P62, 57R19, 32Q60; Secondary: 53C26.

Keywords: Nearly Kähler manifold, special holonomy,

$\mathrm{G}_2$ -manifold, Joyce examples, Kummer construction, formality, rational homotopy type, cohomology algebra, intersection homology.

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