On the formality of nearly Kähler manifolds and of Joyce's examples in $G_2$-holonomy
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| Format: | Preprint |
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2020
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| author | Amann, Manuel Taimanov, Iskander A. |
| author_facet | Amann, Manuel Taimanov, Iskander A. |
| contents | It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already -- an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kähler manifolds. A connection between these two results can be found in the fact that both "special holonomy" and "nearly Kähler" naturally generalize compact Kähler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_10915 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the formality of nearly Kähler manifolds and of Joyce's examples in $G_2$-holonomy Amann, Manuel Taimanov, Iskander A. Differential Geometry Algebraic Topology 53C29, 55P62, 57R19, 32Q60 (Primary), 53C26 (Secondary) It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already -- an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kähler manifolds. A connection between these two results can be found in the fact that both "special holonomy" and "nearly Kähler" naturally generalize compact Kähler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan. |
| title | On the formality of nearly Kähler manifolds and of Joyce's examples in $G_2$-holonomy |
| topic | Differential Geometry Algebraic Topology 53C29, 55P62, 57R19, 32Q60 (Primary), 53C26 (Secondary) |
| url | https://arxiv.org/abs/2012.10915 |