Editing Special unitary group (section)

=== Diffeomorphism with the 3-sphere ''S''<sup>3</sup> ===
If we consider <math>\alpha,\beta</math> as a pair in <math>\mathbb{C}^2</math> where <math>\alpha=a+bi</math> and <math>\beta=c+di</math>, then the equation <math>|\alpha|^2 + |\beta|^2 = 1</math> becomes

<math display="block"> a^2 + b^2 + c^2 + d^2 = 1 </math>

This is the equation of the [[3-sphere|3-sphere S<sup>3</sup>]]. This can also be seen using an embedding: the map

<math display="block">\begin{align}
  \varphi \colon \mathbb{C}^2 \to{} &\operatorname{M}(2, \mathbb{C}) \\[5pt]
         \varphi(\alpha, \beta) ={} &\begin{pmatrix} \alpha & -\overline{\beta}\\ \beta & \overline{\alpha}\end{pmatrix},
\end{align}</math>

where <math>\operatorname{M}(2,\mathbb{C})</math> denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering <math>\mathbb{C}^2</math> [[diffeomorphism|diffeomorphic]] to <math>\mathbb{R}^4</math> and <math>\operatorname{M}(2,\mathbb{C})</math> diffeomorphic to <math>\mathbb{R}^8</math>). Hence, the [[restriction (mathematics)|restriction]] of {{math|''φ''}} to the [[3-sphere]] (since modulus is 1), denoted {{math|''S''<sup>3</sup>}}, is an embedding of the 3-sphere onto a compact submanifold of <math>\operatorname{M}(2,\mathbb{C})</math>, namely {{math|1=''φ''(''S''<sup>3</sup>) = SU(2)}}.

Therefore, as a manifold, {{math|''S''<sup>3</sup>}} is diffeomorphic to {{math|SU(2)}}, which shows that  {{math|SU(2)}} is [[simply connected space|simply connected]] and that {{math|''S''<sup>3</sup>}} can be endowed with the structure of a compact, connected [[Lie group]].