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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 19th 2019

Let $X$ be a closed smooth manifold of dimension 8 with Spin structure. If the frame bundle moreover admits G-structure for

$G = Spin(7) \hookrightarrow Spin(8)$

then the Euler class $\chi$, the second Pontryagin class $p_2$ and the cup product-square $(p_1)^2$ of the first Pontryagin class of the frame bundle/tangent bundle are related by

$8 \chi \;=\; 4 p_2 - (p_1)^2 \,.$
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 15th 2020

• Robert Bryant, Metrics with Exceptional Holonomy, Annals of Mathematics Second Series, Vol. 126, No. 3 (Nov., 1987), pp. 525-576 (jstor:1971360)
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 15th 2020

• Dominic Joyce, A new construction of compact 8-manifolds with holonomy $Spin(7)$, J. Differential Geom. Volume 53, Number 1 (1999), 89-130 (euclid:jdg/1214425448)
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 15th 2020