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This should have an entry of its own (as it does also on Wikipedia) for ease of linking with triality etc.
added pointer to
for discussion of the cohomology of $B Spin(8)$
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 $H^\bullet(B Spin(5), \mathbb{Z})$?
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
$\begin{aligned} \rho_2\left( \tfrac{1}{2}p_1 \right) & = w_4 \\ \rho_2\big( \chi\big) & = w_8 \end{aligned}$I have added pointer to
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 $Spin(5)$?!
added pointers to
Robert Bryant, Remarks on Spinors in Low Dimension (pdf, BryantRemarksOnSpinorsInLowDimension.pdf:file)
Megan M. Kerr, New examples of homogeneous Einstein metrics, Michigan Math. J. Volume 45, Issue 1 (1998), 115-134 (euclid:1030132086)
also pointer to
I have added (here and here, respectively) the statements that
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;
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.
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:
$\big( B tri \big)^\ast \;\colon\; \begin{aligned} \tfrac{1}{2} p_1 & \mapsto \tfrac{1}{2} p_1 \\ \chi & \mapsto \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 + \tfrac{1}{2}\chi \\ \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2}\chi & \mapsto - \big( \tfrac{1}{4}p_2 - \big(\tfrac{1}{4}p_1\big)^2 - \tfrac{1}{2}\chi \big) \end{aligned}$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 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{120}\oplus\mathbb{Z}_{2520}$ | $\mathbb{Z}_2^{\oplus 7}$ |
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