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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexin order to have a good place to record the diagram: $$ \array{ ( q_1, q_2 ) &\mapsto& (x \mapsto q_1 \cdot x \cdot \overline{q}_2) \\ Sp(1) \times Sp(1) &\overset{\simeq}{\longrightarrow}& Spin(4) \\ \big\downarrow && \big\downarrow \\ Sp(1)\cdot Sp(1) &\overset{\simeq}{\longrightarrow}& SO(4) } $$ <a href="https://ncatlab.org/nlab/revision/SO%284%29/1">v1</a>, <a href="https://ncatlab.org/nlab/show/SO%284%29">current</a>

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

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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 8th 2019
   • (edited Apr 8th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexjust for completeness, I added this statement: *** The [[integral cohomology]] [[cohomology ring|ring]] of the [[classifying space]] $B SO(4)$ is $$ H^\bullet \big( p_1, \chi, W_3 \big) / \big( 2 W_3 \big) $$ where * $p_1$ is the [[first Pontryagin class]] * $\chi$ is the [[Euler class]], * $W_3$ is the [[integral Stiefel-Whitney class]]. Notice that the [[cup product]] of the [[Euler class]] with itself is the [[second Pontryagin class]] $$ \chi \smile \chi \;=\; p_2 \,, $$ which therefore, while present, does not appear as a separate generator. *** I hope I got this right that $W_5$ does not appear. <a href="https://ncatlab.org/nlab/revision/diff/SO%284%29/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/SO%284%29/4">v4</a>, <a href="https://ncatlab.org/nlab/show/SO%284%29">current</a>

   just for completeness, I added this statement:

   ---

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

   $$H^\bullet \big( p_1, \chi, W_3 \big) / \big( 2 W_3 \big)$$

   where

   • $p_1$ is the first Pontryagin class

   • $\chi$ is the Euler class,

   • $W_3$ is the integral Stiefel-Whitney class.

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

   $$\chi \smile \chi \;=\; p_2 \,,$$

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

   ---

   I hope I got this right that $W_5$ does not appear.

   diff, v4, current

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 8th 2019
   • (edited Apr 8th 2019)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexYes, it seems to me that you only have $W_3$ out of the integral SW classes, based on one of those sources I sent you.

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 5th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexcopied over the homotopy groups of $SO(4)$ in low degree | $G$ | $\pi_1$ | $\pi_2$ | $\pi_3$ | $\pi_4$ | $\pi_5$ | $\pi_6$ | $\pi_7$ | $\pi_8$ | $\pi_9$ | $\pi_10$ | $\pi_11$ | $\pi_12$ | $\pi_13$ | $\pi_14$ | $\pi_15$ | |--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--| | $SO(4)$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{3}^{\oplus 2}$ | $\mathbb{Z}_{15}^{\oplus 2}$|$\mathbb{Z}_{2}^{\oplus 2}$ |$\mathbb{Z}_{2}^{\oplus 4}$ | $\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}$ | <a href="https://ncatlab.org/nlab/revision/diff/SO%284%29/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/SO%284%29/7">v7</a>, <a href="https://ncatlab.org/nlab/show/SO%284%29">current</a>

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

   $G$	$\pi_1$	$\pi_2$	$\pi_3$	$\pi_4$	$\pi_5$	$\pi_6$	$\pi_7$	$\pi_8$	$\pi_9$	$\pi_10$	$\pi_11$	$\pi_12$	$\pi_13$	$\pi_14$	$\pi_15$
   $SO(4)$	$\mathbb{Z}_2$	0	$\mathbb{Z}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{12}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{3}^{\oplus 2}$	$\mathbb{Z}_{15}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 4}$	$\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2}$	$\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2}$	$\mathbb{Z}_2^{\oplus 4}$

   diff, v7, current