Compact holonomy $\mathrm{G}_2$ manifolds need not be formal

TL;DR

This paper constructs a specific compact G2 manifold that is simply connected and non-formal, demonstrating that such manifolds can have complex topological properties.

Contribution

It provides the first known example of a compact, simply connected G2 manifold that is non-formal, using a novel arrangement of the singular locus.

Findings

Constructed a non-formal, simply connected G2 manifold.

Used a specific configuration of the singular locus to obtain a non-vanishing Massey product.

Demonstrated that compact G2 manifolds need not be formal.

Abstract

We construct a compact, simply connected manifold with holonomy $\mathrm{G}_2$ that is non-formal. We use the construction method of compact torsion-free $\mathrm{G}_2$ manifolds developed by D.D. Joyce and S. Karigiannis. A non-vanishing triple Massey product is obtained by arranging the singular locus in a particular configuration.