Contents

group theory

# Contents

## 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

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}$

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

## References

• 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)