LAUSR

We study a cohomology theory $$H^{\bullet }_{\varphi }$$Hφ∙, which we call the $${\mathcal {L}}_B$$LB-cohomology, on compact torsion-free $$\mathrm {G}_2$$G2 manifolds. We show that $$H^k_{\varphi } \cong H^k_{\mathrm {dR}}$$Hφk≅HdRk for $$k \ne 3, 4$$k≠3,4, but that $$H^k_{\varphi }$$Hφk is infinite-dimensional for $$k = 3,4$$k=3,4. Nevertheless, there is a canonical injection $$H^k_{\mathrm {dR}} \rightarrow H^k_{\varphi }$$HdRk→Hφk. The $${\mathcal {L}}_B$$LB-cohomology also satisfies a Poincaré duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $$\mathrm {d}$$d and the derivation $${\mathcal {L}}_B$$LB and uses both Hodge theory and the special properties of $$\mathrm {G}_2$$G2-structures in an essential way. As an application of our results, we prove that compact torsion-free $$\mathrm {G}_2$$G2 manifolds are ‘almost formal’ in the sense that most of the Massey triple products necessarily must vanish.