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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)

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2019

    added the following statement; which appears as Lemma 2.1 in


    Let

    S 4 BSpin(4) π BSpin(5) \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 4Spin(5)/Spin(4)S^4 \simeq Spin(5)/Spin(4) (this Prop.).

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

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

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 20th 2019

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


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

    1. the first fractional Pontryagin class 12p 1\tfrac{1}{2}p_1

    2. the linear combination 12p 212(p 1) 2\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 (BSpin(5),)[12p 1,12p 212(p 1) 2] 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

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeJan 16th 2021

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

    diff, v15, current

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 16th 2021

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

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

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