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We study a cohomology theory $H^{\bullet}_{\varphi}$, called the $\mathcalL_B$-cohomology, on compact torsion-free $\mathrm{G}_2$-manifolds. We show that$H^k_{\varphi} \cong H^k_{\mathrm{dR}}$ for $k \neq 3, 4$, but that$H^k_{\varphi}$ is infinite-dimensional for $k = 3,4$. Nevertheless there is acanonical injection $H^k_{\mathrm{dR}} \to H^k_{\varphi}$. The $\mathcalL_B$-cohomology also satisfies a Poincar\'e duality induced by the Hodge star.The establishment of these results requires a delicate analysis of theinterplay 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 ourresults, we prove that compact torsion-free $\mathrm{G}_2$-manifolds are'almost formal' in the sense that most of the Massey triple productsnecessarily must vanish.

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