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I have added pointer to
to the entries 7-sphere, ADE classification, Freund-Rubin compactification.
This article proves the neat result that the finite subgroups $\Gamma$ of $SO(8)$ such that $S^7/\Gamma$ is smooth and spin and has at least four Killing spinors has an ADE classification. The $\Gamma$s are the the “binary” versions of the symmetries of the Platonic solids.
I added some more material to 7-sphere, mostly on exotic smooth structures.
I mentioned that these exotic spheres are called Brieskorn manifolds or spheres and corrected a typo ($z^{42}$ – !)
Added statement of the canonical (nearly parallel) $G_2$-structure on the 7-sphere – here.
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