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I am interested in learning about $G_2$ manifolds and am aware that one of the canonical references is Joyce's Compact Manifolds with Special Holonomy. I am certain that my background is, at this moment, not enough to read this research monograph so I would like to know if there are books that can bring me up to speed. My background is in Lee and Do Carmo level.

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  • $\begingroup$ There are currently ongoing efforts on the classification of G2 manifolds. These slides seem to give a nice introductory overview of the subject. Here are a few papers on new invariants and the likes. From here googling the authors should bring up more resources and eventually some books. $\endgroup$ – Ali Caglayan Jan 8 '17 at 15:15
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Joyce's book Riemannian Holonomy Groups and Calibrated Geometry is an extended version of the research monograph you are reading, with more details and background material, aimed at providing a graduate course in the subject.

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Salamon's Riemannian geometry and holonomy groups is another nice place to start.

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I started with this 58 page set of notes by Joyce https://arxiv.org/pdf/math/0108088.pdf, it was published as the first section of a three section book "Calabi-Yau Manifolds and Related Geometries" (the other two sections were written by Gross and Huybrechts). This is lighter than his textbooks.

I also recommend this introductory talk of Robert Bryant https://www.youtube.com/watch?v=j0qBv62pNWw.

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