nLab SO(4)

Idea

The special orthogonal group in dimension 4.

Properties

Exceptional isomorphisms

Proposition

There is a commuting diagram of Lie groups of the form

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

where

1. in the top left we have Sp(1) = Spin(3),

2. in the top right we have Spin(4),

3. in the bottom left we have Sp(1).Sp(1)

4. in the bottom right we have SO(4)

5. the horizontal morphism assigns the conjugation action of unit quaternions, as indicated,

6. the right vertical morphism is the defining double cover,

7. the left vertical morphism is the defining quotient group-projection.

Cohomology

Proposition

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.

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.

Homotopy groups

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	$\mathbb{Z}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{12}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{3}^{\oplus 2}$	$\mathbb{Z}_{15}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 4}$	$\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2}$	$\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2}$	$\mathbb{Z}_2^{\oplus 4}$

Related concepts

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

References

• 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

• Wikipedia, Rotations in 4-dimensional Euclidean space

On the integral cohomology of the classifying space:

• Edgar H. Brown, The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients, Proceedings of the American Mathematical Society, Vol. 85, No. 2 (Jun., 1982), pp. 283-288 (jstor:2044298)

reviewed in

• Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)