---
date: 2021-04-26
---

Tags: #homotopy #physics 

# Homotopy Groups of SO(n)

![Homotopy Groups of $SO^n$](attachments/2-22ReadingNotes-591bd%201%201.png)

# Useful Higher Homotopy used in Physics

![Various higher homotopy groups](attachments/2-22ReadingNotes-0ea10%201%201.png)

$\pi_n$ are equal for the following spaces:

- $SO^3$
- $\RP^3$
- $S^3$
- $SU^2$

 (Maybe these are all diffeomorphic)

Also $\pi_n(\RP^n) = \pi_n(S^n)$.

\[
Sp^4 = SU^2 \cross SU^2
.\]

\[
J: \pi_k(SO^n) \to \pi_{n+k} S^n
.\]


# Homotopy of Infinite Grassmannian

![Homotopy of infinite Grassmannian](attachments/2-22ReadingNotes-f759d%201%201.png)

# Misc
- $\pi_1(SL_n(\RR)) = \ZZ\delta_2 + \ZZ_2 \delta_{n\geq 3}$
  See [Lemma 5.3](http://www.math.rice.edu/~andyp/notes/HomotopySpheresLowDimTop.pdf)
- $\pi_1(SO_n(\RR)) = \pi_1(SL_n(\RR))$