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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 10th 2019

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 16th 2019

added the following statement; which appears as Lemma 2.1 in

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:

$\pi_\ast \left( \chi^3 \right) \;=\; 2 p_2 \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,.$
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 20th 2019

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:

$H^\bullet \big( B Spin(5), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \left[ \tfrac{1}{2}p_1, \; \tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2 \right]$
• CommentRowNumber4.
• CommentAuthorJohn Baez
• CommentTimeJan 16th 2021

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

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeJan 16th 2021

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?