2021-04-26_Homotopy_Groups_of_Other_Spaces

Tags: #homotopy #physics

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
  • \(\pi_1(SO_n({\mathbf{R}})) = \pi_1(SL_n({\mathbf{R}}))\)
#homotopy #physics