The $${mathcal {L}}_B$$LB-cohomology on compact torsion-free $$mathrm {G}_2$$G2 manifolds and an application to ‘almost’ formality
The $${mathcal {L}}_B$$LB-cohomology on compact torsion-free $$mathrm {G}_2$$G2 manifolds and an application to ‘almost’ formality
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紧致无扭转 $$mathrm {G}_2$$G2 流形上的 $${mathcal {L}}_B$$LB 上同调以及“几乎”形式的应用
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发表时间：
2018
影响因子：
0.7
通讯作者：
Chi Cheuk Tsang
中科院分区：
文献类型：
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作者：
Ki Fung Chan;Spiro Karigiannis;Chi Cheuk Tsang
We study a cohomology theory $$H^{ullet }_{varphi }$$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 $$k
e 3, 4$$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 $$H^k_{mathrm {dR}}
ightarrow H^k_{varphi }$$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.
We study a cohomology theory $$H^{ullet }_{varphi }$$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 $$k
e 3, 4$$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 $$H^k_{mathrm {dR}}
ightarrow H^k_{varphi }$$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.