fundamental group

\(\pi_1 \operatorname{GL}_n({\mathbf{C}}) = \pi_1{\mathsf{Sp}}_n({\mathbf{R}}) = \pi_1 {\operatorname{U}}_n = {\mathbf{Z}}\)
\(\pi_1 {\operatorname{SO}}_2({\mathbf{R}}) = {\mathbf{Z}}\)
- \(\pi_1 {\operatorname{SO}}_n({\mathbf{R}}) = C_2\) when \(n\geq 3\).
\(\pi_1 {\operatorname{SL}}_n({\mathbf{C}}) = \pi_1 {\operatorname{SL}}_n({\mathbf{C}}) = 1\)
\(\pi_1 {\mathsf{Sp}}_n({\mathbf{C}}) = 1\)
\(\pi_* {\operatorname{O}}_n = [C_2, C_2, 0, {\mathbf{Z}}, 0, 0, 0, {\mathbf{Z}}, \cdots]\)
\(\pi_* {\mathsf{K}}{\operatorname{O}}_n = [{\mathbf{Z}}, C_2, C_2, 0, {\mathbf{Z}}, 0, 0, 0, {\mathbf{Z}}, \cdots]\)

Higher homotopy

\(\pi_{*}(M O)=\mathbb{F}_{2}\left[x_{i}: i+1\right.\) is not a power of 2\(] .\)
\(\Omega_{*}^{S O} \otimes \mathbb{Q}=\mathbb{Q}\left[\left[\mathbf{C} P^{2}\right],\left[\mathbf{C} P^{4}\right], \ldots\right]\)
\(\pi_{*}(M U)=\mathbb{Z}\left[x_{1}, x_{2}, \ldots\right], \quad\left|x_{i}\right|=2 i\)
\(\pi_{*}(B P)=\mathbb{Z}_{(p)}\left[v_{1}, v_{2}, \ldots\right], \quad\left|v_{i}\right|=2 p^{i}-2\)