nLab
SO(4)

Contents

Contents

Idea

The special orthogonal group in dimension 4.

Properties

Exceptional isomorphisms

Proposition

There is a commuting diagram of Lie groups of the form

(q 1,q 2) (xq 1xq¯ 2) Sp(1)×Sp(1) Spin(4) Sp(1)Sp(1) SO(4) \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 BSO(4)B SO(4) is

H (p 1,χ,W 3)/(2W 3) 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

χχ=p 2, \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)SO(4) in low degrees are

GGπ 1\pi_1π 2\pi_2π 3\pi_3π 4\pi_4π 5\pi_5π 6\pi_6π 7\pi_7π 8\pi_8π 9\pi_9π 10\pi_10π 11\pi_11π 12\pi_12π 13\pi_13π 14\pi_14π 15\pi_15
SO(4)SO(4) 2\mathbb{Z}_20 2\mathbb{Z}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 12 2\mathbb{Z}_{12}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 3 2\mathbb{Z}_{3}^{\oplus 2} 15 2\mathbb{Z}_{15}^{\oplus 2} 2 2\mathbb{Z}_{2}^{\oplus 2} 2 4\mathbb{Z}_{2}^{\oplus 4} 2 2 12 2\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2} 2 4 84 2\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2} 2 4\mathbb{Z}_2^{\oplus 4}


rotation groups in low dimensions:

sp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)Pin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)SO(8)
SO(9)Spin(9)
\vdots\vdots
SO(16)Spin(16)SemiSpin(16)
SO(32)Spin(32)SemiSpin(32)

see also

References

  • Jason Hanson, Rotations in three, four, and five dimensions (arXiv:1103.5263)

See also

On the integral cohomology of the classifying space:

  • Edgar H. Brown, The Cohomology of BSO nB SO_n and BO nBO_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)

Last revised on June 5, 2019 at 05:31:53. See the history of this page for a list of all contributions to it.