The ${\mathcal{L}}_B$-cohomology on compact torsion-free ${\mathrm{G}}_2$ manifolds and an application to 'almost' formality

We study a cohomology theory $H_{\phi }^{\bullet }$, called the $\mathcal L_B$-cohomology,oncompacttorsion-free$$\mathrm{G}$2$-manifolds\mathrm{.}Weshowthat$H^k{\varphi} \cong H^k_{$\mathrm{d}\mathrm{R}$}$for$k \neq 3, 4$,butthat$H^k_{\varphi}$isinfinite-dimensionalfor$k = 3,4$. Nevertheless there is a canonical injection $H_{\mathrm{d}\mathrm{R}}^k\to H_{\phi }^k$. The $\mathcal L_B$-cohomology also satisfies a Poincar'e duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $\mathrm{d}$ and the derivation ${\mathcal{L}}_B$, and uses both Hodge theory and the special properties of ${\mathrm{G}}_2$-structures in an essential way. As an application of our results, we prove that compact torsion-free ${\mathrm{G}}_2$-manifolds are ‘almost formal’ in the sense that most of the Massey triple products necessarily must vanish.