Homotopy Groups of SO(n)
Homotopy Groups of SO^n
Useful Higher Homotopy used in Physics
Various higher homotopy groups
\(\pi_n\) are equal for the following spaces:
- \(SO^3\)
- \({\mathbf{RP}}^3\)
- \(S^3\)
- \(SU^2\)
(Maybe these are all diffeomorphic)
Also \(\pi_n({\mathbf{RP}}^n) = \pi_n(S^n)\).
\begin{align*} Sp^4 = SU^2 \times SU^2 .\end{align*}
\begin{align*} J: \pi_k(SO^n) \to \pi_{n+k} S^n .\end{align*}
Homotopy of Infinite Grassmannian
Homotopy of infinite Grassmannian
Misc
- \(\pi_1(SL_n({\mathbf{R}})) = {\mathbf{Z}}\delta_2 + {\mathbf{Z}}_2 \delta_{n\geq 3}\) See Lemma 5.3
- \(\pi_1(SO_n({\mathbf{R}})) = \pi_1(SL_n({\mathbf{R}}))\)