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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

  1. Corrected description of rightmost map in the diagram in Prop. 2.3.

    Robert Bruner

    diff, v21, current

  2. Corrected right most map in the diagram in Prop. 2.3 and its description.

    Robert Bruner

    diff, v21, current