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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 10th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor ease of reference, and to go along with _[[SO(2)]], [[Spin(2)]], [[Pin(2)]], [[Spin(3)]], [[Spin(4)]]_ <a href="https://ncatlab.org/nlab/revision/Spin%285%29/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Spin%285%29">current</a>

   for ease of reference, and to go along with SO(2), Spin(2), Pin(2), Spin(3), Spin(4)

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 16th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the following statement; which appears as Lemma 2.1 in * [[Raoul Bott]], [[Alberto Cattaneo]], _Integral Invariants of 3-Manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001)) *** Let $$ \array{ S^4 &\longrightarrow& B Spin(4) \\ && \big\downarrow^{\mathrlap{\pi}} \\ && B Spin(5) } $$ be the [[spherical fibration]] of [[classifying spaces]] induced from the canonical inclusion of [[Spin(4)]] into [[Spin(5)]] and using that the [[4-sphere]] is equivalently the [[coset space]] $S^4 \simeq Spin(5)/Spin(4)$ ([this Prop.](sphere#nSphereAsCosetSpace)). Then the [[fiber integration]] of the triple [[cup product|cup power]] of the [[Euler class]] $\chi \in H^4\big( B Spin(4), \mathbb{Z}\big)$ (see [this Prop](Spin4#IntegralCohomologyOfClassifyingSpace)) is twice the [[second Pontryagin class]]: $$ \pi_\ast \left( \chi^3 \right) \;=\; 2 p_2 \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,. $$ <a href="https://ncatlab.org/nlab/revision/diff/Spin%285%29/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%285%29/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Spin%285%29">current</a>

   added the following statement; which appears as Lemma 2.1 in

   • Raoul Bott, Alberto Cattaneo, Integral Invariants of 3-Manifolds, J. Diff. Geom., 48 (1998) 91-133 (arXiv:dg-ga/9710001)

   ---

   Let

   $$\array{ S^4 &\longrightarrow& B Spin(4) \\ && \big\downarrow^{\mathrlap{\pi}} \\ && B Spin(5) }$$

   be the spherical fibration of classifying spaces induced from the canonical inclusion of Spin(4) into Spin(5) and using that the 4-sphere is equivalently the coset space $S^4 \simeq Spin(5)/Spin(4)$ (this Prop.).

   Then the fiber integration of the triple cup power of the Euler class $\chi \in H^4\big( B Spin(4), \mathbb{Z}\big)$ (see this Prop) is twice the second Pontryagin class:

   $$\pi_\ast \left( \chi^3 \right) \;=\; 2 p_2 \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,.$$

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 20th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the following statement, but for the moment without good referencing: *** The [[integral cohomology|integral]] [[cohomology ring]] of the [[classifying space]] $B Spin(5)$ is spanned by two generators 1. the [[first fractional Pontryagin class]] $\tfrac{1}{2}p_1$ 1. the [[linear combination]] $\tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2$ of the half the [[second Pontryagin class]] with half the [[cup product]]-square of the [[first Pontryagin class]]: $$ H^\bullet \big( B Spin(5), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \left[ \tfrac{1}{2}p_1, \; \tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2 \right] $$ <a href="https://ncatlab.org/nlab/revision/diff/Spin%285%29/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%285%29/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Spin%285%29">current</a>

   added the following statement, but for the moment without good referencing:

   ---

   The integral cohomology ring of the classifying space $B Spin(5)$ is spanned by two generators

   1. the first fractional Pontryagin class $\tfrac{1}{2}p_1$

   2. the linear combination $\tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2$ of the half the second Pontryagin class with half the cup product-square of the first Pontryagin class:

   $$H^\bullet \big( B Spin(5), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \left[ \tfrac{1}{2}p_1, \; \tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2 \right]$$

   diff, v4, current

4. • CommentRowNumber4.
   • CommentAuthorJohn Baez
   • CommentTimeJan 16th 2021
   • PermaLink
   Author: John Baez
   Format: MarkdownItexAdded a proof that Spin(5) is isomorphic to Sp(2). <a href="https://ncatlab.org/nlab/revision/diff/Spin%285%29/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%285%29/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Spin%285%29">current</a>

   Added a proof that Spin(5) is isomorphic to Sp(2).

   diff, v15, current

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 16th 2021
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex>every invariant subspace has an invariant complement, so one or both of the 6-dimensional subspaces $\Lambda_+^2 V$ and $\Lambda_-^2 V$ must have a 5-dimensional subspace invariant under the action of $\mathrm{Sp}(V)$ I get that the invariant subspace at the start of this quote is generated by the element $J$ preserved by $SU(4)$, but it's not clear what the subspace is. Is it the real span? Why the claim that **one or both** of $\Lambda_+^2 V$ and $\Lambda_-^2 V$ have an invariant subspace?

   every invariant subspace has an invariant complement, so one or both of the 6-dimensional subspaces $\Lambda_+^2 V$ and $\Lambda_-^2 V$ must have a 5-dimensional subspace invariant under the action of $\mathrm{Sp}(V)$

   I get that the invariant subspace at the start of this quote is generated by the element $J$ preserved by $SU(4)$, but it’s not clear what the subspace is. Is it the real span? Why the claim that one or both of $\Lambda_+^2 V$ and $\Lambda_-^2 V$ have an invariant subspace?