---
date: 2022-04-08 19:56
modification date: Friday 8th April 2022 19:56:23
title: "fundamental group"
aliases: [fundamental group]
---

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

---
- Tags: 
	- #examples
- Refs:
	- #todo/add-references
- Links:
	- [examples of fibrations](Unsorted/list%20of%20fibrations.md)
	- [examples of cohomology rings](Unsorted/examples%20of%20cohomology%20rings.md)

---

# fundamental group

- $\pi_1 \GL_n(\CC) = \pi_1\Sp_n(\RR) = \pi_1 \U_n =  \ZZ$
- $\pi_1 \SO_2(\RR) = \ZZ$
	- $\pi_1 \SO_n(\RR) = C_2$ when $n\geq 3$.
- $\pi_1 \SL_n(\CC) = \pi_1 \SL_n(\CC) = 1$
- $\pi_1 \Sp_n(\CC) = 1$
- $\pi_* \Orth_n = [C_2, C_2, 0, \ZZ, 0, 0, 0, \ZZ, \cdots]$
- $\pi_* \K \Orth_n = [\ZZ, C_2, C_2, 0, \ZZ, 0, 0, 0, \ZZ, \cdots]$

# Higher homotopy


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