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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
    • CommentTime4 days ago

    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

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