Abstract
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 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.
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Acknowledgements
These results were obtained in 2017 as part of the collaboration between the COSINE program organized by the Chinese University of Hong Kong and the URA program organized by the University of Waterloo. The authors thank both universities for this opportunity. Part of the writing was done, while the second author held a Fields Research Fellowship at the Fields Institute. The second author thanks the Fields Institute for their hospitality. The authors also thank the anonymous referee for pointing out that we had actually also established Theorem 4.10, which is the stronger version of Corollary 4.9 in the case of full \(\mathrm {G}_2\) holonomy.
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Chan, K.F., Karigiannis, S. & Tsang, C.C. The \({\mathcal {L}}_B\)-cohomology on compact torsion-free \(\mathrm {G}_2\) manifolds and an application to ‘almost’ formality. Ann Glob Anal Geom 55, 325–369 (2019). https://doi.org/10.1007/s10455-018-9629-x
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DOI: https://doi.org/10.1007/s10455-018-9629-x