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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2019

    in order to have a good place to record the diagram:

    (q 1,q 2) (xq 1xq¯ 2) Sp(1)×Sp(1) Spin(4) Sp(1)Sp(1) SO(4) \array{ ( q_1, q_2 ) &\mapsto& (x \mapsto q_1 \cdot x \cdot \overline{q}_2) \\ Sp(1) \times Sp(1) &\overset{\simeq}{\longrightarrow}& Spin(4) \\ \big\downarrow && \big\downarrow \\ Sp(1)\cdot Sp(1) &\overset{\simeq}{\longrightarrow}& SO(4) }

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2019
    • (edited Apr 8th 2019)

    just for completeness, I added this statement:

    The integral cohomology ring of the classifying space BSO(4)B SO(4) is

    H (p 1,χ,W 3)/(2W 3) H^\bullet \big( p_1, \chi, W_3 \big) / \big( 2 W_3 \big)


    Notice that the cup product of the Euler class with itself is the second Pontryagin class

    χχ=p 2, \chi \smile \chi \;=\; p_2 \,,

    which therefore, while present, does not appear as a separate generator.

    I hope I got this right that W 5W_5 does not appear.

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 8th 2019
    • (edited Apr 8th 2019)

    Yes, it seems to me that you only have W 3W_3 out of the integral SW classes, based on one of those sources I sent you.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2019

    copied over the homotopy groups of SO(4)SO(4) in low degree

    GG π 1\pi_1 π 2\pi_2 π 3\pi_3 π 4\pi_4 π 5\pi_5 π 6\pi_6 π 7\pi_7 π 8\pi_8 π 9\pi_9 π 10\pi_10 π 11\pi_11 π 12\pi_12 π 13\pi_13 π 14\pi_14 π 15\pi_15
    SO(4)SO(4) 2\mathbb{Z}_2 0 2\mathbb{Z}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 12 2\mathbb{Z}_{12}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 3 2\mathbb{Z}_{3}^{\oplus 2} 15 2\mathbb{Z}_{15}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 2 4\mathbb{Z}_{2}^{\oplus 4} 2 2 12 2\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2} 2 4 84 2\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2} 2 4\mathbb{Z}_2^{\oplus 4}

    diff, v7, current