The special orthogonal group in dimension 4.
There is a commuting diagram of Lie groups of the form
where
in the top right we have Spin(4),
in the bottom left we have Sp(1).Sp(1)
in the bottom right we have SO(4)
the horizontal morphism assigns the conjugation action of unit quaternions, as indicated,
the right vertical morphism is the defining double cover,
the left vertical morphism is the defining quotient group-projection.
The integral cohomology ring of the classifying space $B SO(4)$ is
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
which therefore, while present, does not appear as a separate generator.
This is a special case of Brown 82, Theorem 1.5, reviewed for instance as Rudolph-Schmidt 17, Thm. 4.2.23 with Rmk. 4.2.25.
The homotopy groups of $SO(4)$ in low degrees are
$G$ | $\pi_1$ | $\pi_2$ | $\pi_3$ | $\pi_4$ | $\pi_5$ | $\pi_6$ | $\pi_7$ | $\pi_8$ | $\pi_9$ | $\pi_10$ | $\pi_11$ | $\pi_12$ | $\pi_13$ | $\pi_14$ | $\pi_15$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
$SO(4)$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{3}^{\oplus 2}$ | $\mathbb{Z}_{15}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 4}$ | $\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}$ |
rotation groups in low dimensions:
see also
Marcel Berger, Section 8.9.8 of: Geometry I, Springer 1987 (doi:10.1007/978-3-540-93815-6)
Jason Hanson, Rotations in three, four, and five dimensions (arXiv:1103.5263)
See also
On the integral cohomology of the classifying space:
reviewed in
Last revised on June 19, 2021 at 10:14:59. See the history of this page for a list of all contributions to it.