The $${\mathcal {L}}_B$$ L B -cohomology on compact torsion-free $$\mathrm {G}_2$$ G 2 manifolds and an application to ‘almost’ formality

We study a cohomology theory $$H^{\bullet }_{\varphi }$$ , which we call the $${\mathcal {L}}_B$$ -cohomology, on compact torsion-free $$\mathrm {G}_2$$ manifolds. We show that $$H^k_{\varphi } \cong H^k_{\mathrm {dR}}$$ for $$k \ne 3, 4$$ , but that $$H^k_{\varphi }$$ is infinite-dimensional for $$k = 3,4$$ . Nevertheless, there is a canonical injection $$H^k_{\mathrm {dR}} \rightarrow H^k_{\varphi }$$ . The $${\mathcal {L}}_B$$ -cohomology also satisfies a Poincare 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.