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:

$SO(n,\mathbb{R})=\{A\in {\mathrm{M}}_{\mathrm{a}}(\mathbb{R}):\mathrm{d}\mathrm{e}\mathrm{t}A=1,\mathrm{a}\mathrm{n}\mathrm{d}{A}^T=A^{-1}\}$

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