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

Abstract

We study a cohomology theory $H_{\phi }^{\bullet }$, called the ${\mathcal{L}}_B$-cohomology, on compact torsion-free ${\mathrm{G}}_2$-manifolds. We show that $H_{\phi }^k\cong H_{\mathrm{dR}}^k$ for $k\ne 3,4$, but that $H_{\phi }^k$ is infinite-dimensional for $k=3,4$. Nevertheless there is a canonical injection $H_{\mathrm{dR}}^k\to H_{\phi }^k$. The ${\mathcal{L}}_B$-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 $\mathrm{d}$ 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.

43 pages, 8 figures in colour. Version 3: Minor revisions following referee's report. Main result strengthened in the case of full G2 holonomy. Final version, to appear in Annals of Global Analysis and Geometry