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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 19th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded this statement: *** Let $X$ be a [[closed manifold|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 \[ 8 \chi \;=\; 4 p_2 - (p_1)^2 \,. \] <a href="https://ncatlab.org/nlab/revision/diff/Spin%287%29+manifold/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%287%29+manifold/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Spin%287%29+manifold">current</a>

   added this statement:

   ---

   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

   $$8 \chi \;=\; 4 p_2 - (p_1)^2 \,.$$

   diff, v5, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 15th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Robert Bryant]], _Metrics with Exceptional Holonomy_, Annals of Mathematics Second Series, Vol. 126, No. 3 (Nov., 1987), pp. 525-576 ([jstor:1971360](https://www.jstor.org/stable/1971360)) <a href="https://ncatlab.org/nlab/revision/diff/Spin%287%29+manifold/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%287%29+manifold/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Spin%287%29+manifold">current</a>

   added pointer to

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

   diff, v15, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 15th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[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](https://projecteuclid.org/euclid.jdg/1214425448)) <a href="https://ncatlab.org/nlab/revision/diff/Spin%287%29+manifold/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%287%29+manifold/16">v16</a>, <a href="https://ncatlab.org/nlab/show/Spin%287%29+manifold">current</a>

   added pointer to

   • 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)

   diff, v16, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 15th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * Ya. V. Bazaikin, _On the new examples of complete noncompact Spin(7)-holonomy metrics_, Sib Math J **48**, 8–25 (2007) ([doi:10.1007/s11202-007-0003-7](https://doi.org/10.1007/s11202-007-0003-7)) <a href="https://ncatlab.org/nlab/revision/diff/Spin%287%29+manifold/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%287%29+manifold/18">v18</a>, <a href="https://ncatlab.org/nlab/show/Spin%287%29+manifold">current</a>

   added pointer to

   • Ya. V. Bazaikin, On the new examples of complete noncompact Spin(7)-holonomy metrics, Sib Math J 48, 8–25 (2007) (doi:10.1007/s11202-007-0003-7)

   diff, v18, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 16th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], Thm. 10.7 in: _Spin geometry_, Princeton University Press (1989) &lbrack;[ISBN 9780691085425](https://press.princeton.edu/books/hardcover/9780691085425/spin-geometry-pms-38-volume-38)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Spin%287%29+manifold/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%287%29+manifold/20">v20</a>, <a href="https://ncatlab.org/nlab/show/Spin%287%29+manifold">current</a>

   added pointer to:

   • H. Blaine Lawson, Marie-Louise Michelsohn, Thm. 10.7 in: Spin geometry, Princeton University Press (1989) [ISBN 9780691085425]

   diff, v20, current