---
date: 2022-04-16 20:39
modification date: Saturday 16th April 2022 20:39:01
title: "examples of homotopy groups"
aliases: [examples of homotopy groups]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [examples of homotopy groups of spectra](examples%20of%20homotopy%20groups%20of%20spectra.md)

---

# examples of homotopy groups


- $\pi_1 S^1 = \ZZ$
- $\pi_1 X = C_2$ for $X =\RP^n, n\geq 2$, or $X=\SO_n(\RR)$ for $n\geq 3$
- $\pi_1 X = 1$ for $X = \SU_n(\CC), \SL_n(\CC), \SP_{2n}(\CC)$.
- $\pi^1((S^1)\cartpower n) = \ZZ\cartpower n$
- $\pi_1((S^1)^{\bigvee^n}) = \ZZ^{\ast^n}$

# General properties

- $\pi_1 \bigvee_i X_i = \bigast_n \pi_1 X_i$
- $\pi_1 \prod_i X_i = \prod_i \pi_1 X_i$