examples of homotopy groups

$\pi_1 S^1 = {\mathbf{Z}}$
$\pi_1 X = C_2$ for $X ={\mathbf{RP}}^n, n\geq 2$, or $X={\operatorname{SO}}_n({\mathbf{R}})$ for $n\geq 3$
$\pi_1 X = 1$ for $X = {\operatorname{SU}}_n({\mathbf{C}}), {\operatorname{SL}}_n({\mathbf{C}}), \SP_{2n}({\mathbf{C}})$.
$\pi^1((S^1){ {}^{ \scriptscriptstyle\times^{n} } }) = {\mathbf{Z}}{ {}^{ \scriptscriptstyle\times^{n} } }$
$\pi_1((S^1)^{\bigvee^n}) = {\mathbf{Z}}^{\ast^n}$

General properties