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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2018

    Page created, but author did not leave any comments.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2018

    This should have an entry of its own (as it does also on Wikipedia) for ease of linking with triality etc.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 20th 2019
    • (edited Mar 20th 2019)

    added pointer to

    • Alfred Gray, Paul S. Green, Sphere transitive structures and the triality automorphism, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (euclid:1102976640)

    for discussion of the cohomology of BSpin(8)B Spin(8)

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 20th 2019

    I was really looking for answer to the following question, but not sure yet:

    What is the restriction of the Euler class on BSpin(8)B Spin(8) to BSpin(5)B Spin(5), expressed in terms of the generators of H (BSpin(5),)H^\bullet(B Spin(5), \mathbb{Z})?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2019

    added statement of the cohomology ring of BSpin(8)B Spin(8):


    The ordinary cohomology ring of the classifying space BSpin(8)B Spin(8) is:

    1) with coefficients in the cyclic group of order 2:

    H (BSpin(8), 2)[w 4,w 6,w 7,w 8,ρ 2(14(p 2(12p 1) 22χ))] H^\bullet \big( B Spin(8), \mathbb{Z}_2 \big) \;\simeq\; \mathbb{Z} \big[ w_4, w_6, w_7, w_8, \; \rho_2 \left( \tfrac{1}{4} \left( p_2 - \big(\tfrac{1}{2}p_1\big)^2 - 2 \chi \right) \right) \big]

    where w iw_i are the universal Stiefel-Whitney classes,

    and where

    ρ 2:H (BSpin(8),)H (BSpin(8), 2) \rho_2 \;\colon\; H^\bullet(B Spin(8), \mathbb{Z}) \to H^\bullet(B Spin(8), \mathbb{Z}_2)

    is mod 2 reduction

    2) with coefficients in the integers:

    H (BSpin(8),)[12p 1,14(p 2(12p 1) 22χ),χ,δw 6]/2δw 6, H^\bullet \big( B Spin(8), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \Big[ \tfrac{1}{2}p_1, \; \tfrac{1}{4} \left( p_2 - \big(\tfrac{1}{2}p_1\big)^2 - 2 \chi \right) , \; \chi, \; \delta w_6 \Big] / \big\langle 2 \delta w_6\big\rangle \,,

    where p 1p_1 is the first fractional Pontryagin class, p 2p_2 is the second Pontryagin class, χ\chi is the Euler class.

    Moreover, we have the relations

    ρ 2(12p 1) =w 4 ρ 2(χ) =w 8 \begin{aligned} \rho_2\left( \tfrac{1}{2}p_1 \right) & = w_4 \\ \rho_2\big( \chi\big) & = w_8 \end{aligned}

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2019

    I have added pointer to

    and added two of its Propositions to the text here, expanding on those distinct Spin(7)Spin(7)-subgroups of Spin(8)Spin(8) intersecting in G 2G_2.

    Now I have a question: By that sequence of pulback squares we should get analogous distinct subgroup inclusions (in distinct conjugacy classes) of SU(3)SU(3) in Spin(6)Spin(6) and of SU(2)SU(2) in Spin(5)Spin(5)?!

    diff, v8, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2019

    added pointers to

    diff, v10, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2019

    also pointer to

    • Andreas Kollross, Prop. 3.3 of A Classification of Hyperpolar and Cohomogeneity One Actions, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 (jstor:2693761)

    diff, v10, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2019
    • (edited Apr 10th 2019)

    I have added (here and here, respectively) the statements that

    1. The three groups

      • Sp(1)Sp(2)Sp(1)\cdot Sp(2),

      • Sp(2)Sp(1)Sp(2)\cdot Sp(1),

      • SO(3)×SO(5)SO(3) \times SO(5)

      with their canonical embeddings into SO(8)SO(8) represent three distinct conjugacy classes of subgroups of SO(8)SO(8) and triality permutes these three transitively;

    2. The three groups

      • Sp(1)Sp(2)Sp(1)\cdot Sp(2),

      • Sp(2)Sp(1)Sp(2)\cdot Sp(1),

      • Spin(3)Spin(5)Spin(3)\cdot Spin(5)

      with their canonical embeddings into Spin(8)Spin(8) represent three distinct conjugacy classes of subgroups of Spin(8)Spin(8) and triality permutes these three transitively;

    Hope I got this right. This is pretty subtle.

    diff, v11, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2019

    added a summary xymatrix-diagram here. But it comes out too large.

    (From error messages I see that the xymatrix gets passed an @=5em, that’s maybe too much.)

    diff, v12, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2019
    • (edited Apr 28th 2019)

    added statement of the action of triality on the universal characteristic classes, paraphrasing Čadek-Vanžura 97, Lemma 4.2:


    Consider the delooping of the triality automorphism relating Sp(2).Sp(1) with Spin(5).Spin(3), from above, on classifying spaces

    B(Spin(5)Spin(3)) BSpin(8) Btri B(Sp(2)Sp(1)) BSpin(8) \array{ B \big( Spin(5) \cdot Spin(3) \big) &\hookrightarrow& B Spin(8) \\ \big\downarrow && \big\downarrow^{ B \mathrlap{tri} } \\ B \big( Sp(2) \cdot Sp(1) \big) &\hookrightarrow& B Spin(8) }

    Then the pullback of the universal characteristic classes of BSpin(8)B Spin(8) (from above) along BtriB tri is as follows:

    (Btri) *:12p 1 12p 1 χ 14p 2(14p 1) 2+12χ 14p 2(14p 1) 212χ (14p 2(14p 1) 212χ) \big( B tri \big)^\ast \;\colon\; \begin{aligned} \tfrac{1}{2} p_1 & \mapsto \tfrac{1}{2} p_1 \\ \chi & \mapsto \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 + \tfrac{1}{2}\chi \\ \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2}\chi & \mapsto - \big( \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2}\chi \big) \end{aligned}

    diff, v13, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2019
    • (edited Jun 5th 2019)

    copied over the homotopy groups in low degrees:

    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(8)SO(8) 2\mathbb{Z}_2 00 \mathbb{Z} 00 00 00 2\mathbb{Z}^{\oplus 2} 2 3\mathbb{Z}_{2}^{\oplus 3} 2 3\mathbb{Z}_{2}^{\oplus 3} 8 24\mathbb{Z}_{8} \oplus \mathbb{Z}_{24} 2\mathbb{Z}_2 \oplus \mathbb{Z} 0 2\mathbb{Z}^{\oplus 2} 2 8 120 2520\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{120}\oplus\mathbb{Z}_{2520} 2 7\mathbb{Z}_2^{\oplus 7}

    diff, v17, current

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