The \(\mathcal{L}_B\)-cohomology on compact torsion-free \(\mathrm{G}_2\) manifolds and an application to ‘almost’ formality. (English) Zbl 1505.53036
Summary: 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 \(k \neq 3, 4\), 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 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.
MSC:
| 53C10 | \(G\)-structures |
| 57R19 | Algebraic topology on manifolds and differential topology |
| 53C29 | Issues of holonomy in differential geometry |
| 58A12 | de Rham theory in global analysis |
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