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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 2nd 2018

Page created, but author did not leave any comments.

• 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)

• 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 $B Spin(8)$

• 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 $B Spin(8)$ to $B Spin(5)$, expressed in terms of the generators of $H^\bullet(B Spin(5), \mathbb{Z})$?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 22nd 2019

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

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

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

$H^\bullet \big( B Spin(8), \mathbb{Z}_2 \big) \;\simeq\; \mathbb{Z} \big[ w_4, w_6, w_7, w_8, \; \rho_2 \left( \tfrac{1}{4} \left( p_2 - \big(\tfrac{1}{2}p_1\big)^2 - 2 \chi \right) \right) \big]$

where $w_i$ are the universal Stiefel-Whitney classes,

and where

$\rho_2 \;\colon\; H^\bullet(B Spin(8), \mathbb{Z}) \to H^\bullet(B Spin(8), \mathbb{Z}_2)$

2) with coefficients in the integers:

$H^\bullet \big( B Spin(8), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \Big[ \tfrac{1}{2}p_1, \; \tfrac{1}{4} \left( p_2 - \big(\tfrac{1}{2}p_1\big)^2 - 2 \chi \right) , \; \chi, \; \delta w_6 \Big] / \big\langle 2 \delta w_6\big\rangle \,,$

where $p_1$ is the first fractional Pontryagin class, $p_2$ is the second Pontryagin class, $\chi$ is the Euler class.

Moreover, we have the relations

\begin{aligned} \rho_2\left( \tfrac{1}{2}p_1 \right) & = w_4 \\ \rho_2\big( \chi\big) & = w_8 \end{aligned}
• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 30th 2019

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

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

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 10th 2019

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

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

1. The three groups

• $Sp(1)\cdot Sp(2)$,

• $Sp(2)\cdot Sp(1)$,

• $SO(3) \times SO(5)$

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

2. The three groups

• $Sp(1)\cdot Sp(2)$,

• $Sp(2)\cdot Sp(1)$,

• $Spin(3)\cdot Spin(5)$

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

Hope I got this right. This is pretty subtle.

• 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.)

• 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

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

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

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

copied over the homotopy groups in low degrees:

$G$ $\pi_1$ $\pi_2$ $\pi_3$ $\pi_4$ $\pi_5$ $\pi_6$ $\pi_7$ $\pi_8$ $\pi_9$ $\pi_10$ $\pi_11$ $\pi_12$ $\pi_13$ $\pi_14$ $\pi_15$
$SO(8)$ $\mathbb{Z}_2$ $0$ $\mathbb{Z}$ $0$ $0$ $0$ $\mathbb{Z}^{\oplus 2}$ $\mathbb{Z}_{2}^{\oplus 3}$ $\mathbb{Z}_{2}^{\oplus 3}$ $\mathbb{Z}_{8} \oplus \mathbb{Z}_{24}$ $\mathbb{Z}_2 \oplus \mathbb{Z}$ 0 $\mathbb{Z}^{\oplus 2}$ $\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{120}\oplus\mathbb{Z}_{2520}$ $\mathbb{Z}_2^{\oplus 7}$