Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
in order to have a good place to record the diagram:
$\array{ ( q_1, q_2 ) &\mapsto& (x \mapsto q_1 \cdot x \cdot \overline{q}_2) \\ Sp(1) \times Sp(1) &\overset{\simeq}{\longrightarrow}& Spin(4) \\ \big\downarrow && \big\downarrow \\ Sp(1)\cdot Sp(1) &\overset{\simeq}{\longrightarrow}& SO(4) }$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.
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.
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 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{3}^{\oplus 2}$ | $\mathbb{Z}_{15}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 4}$ | $\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}$ |
1 to 4 of 4