Special orthogonal group over reals

Definition

Suppose $n$ is a natural number. The special orthogonal group of degree $n$ over the reals, denoted $SO(n,\mathbb {R} )$, is a Lie group that can be defined concretely as the group of $n\times n$ matrices with real entries whose determinant is 1 and whose product with the transpose is the identity matrix:

$SO(n,\mathbb {R} )=\{A\in \operatorname {Mat} _{n}(\mathbb {R} ):\operatorname {det} A=1,\ \operatorname {and} \ A^{T}=A^{-1}\}$

Particular cases

Value of $n$	Dimension of $SO(n,\mathbb {R} )$ (in general equals $n(n+1)/2-1$)	Description of $SO(n,\mathbb {R} )$, or rather, its underlying topological space
1	0	one-point space
2	1	circle $S^{1}$
3	3	real projective three-dimensional space $\mathbb {P} ^{3}(\mathbb {R} )$
4	6	special orthogonal group:SO(4,R)
5	10	special orthogonal group:SO(5,R)

Algebraic topology

Homology

Further information: homology of special orthogonal group over reals

Cohomology

Further information: cohomology of special orthogonal group over reals

Homotopy

Further information: homotopy of special orthogonal group over reals